Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): O.V. Borodin, A.O. Ivanova, D.V. Nikiforov In 1940, in attempts to solve the Four Color Problem, H. Lebesgue gave an approximate description of the neighborhoods of 5-vertices in the class P 5 of 3-polytopes with minimum degree 5. Given a 3-polytope P , by h ( P ) we denote the minimum of the maximum degrees (height) of the neighborhoods of 5-vertices (minor 5-stars) in P . In 1996, S. Jendrol’ and T. Madaras showed that if a polytope P in P 5 is allowed to have a 5 -vertex adjacent to four 5-vertices, then h ( P ) can be arbitrarily large. For each P without vertices of degree 6 and 5 -vertices adjacent to four 5-vertices in P 5 , it follows from Lebesgue’s Theorem that h ( P ) ≤ 41 . Recently, this bound was lowered to h ( P ) ≤ 28 by O. Borodin, A. Ivanova, and T. Jensen and then to h ( P ) ≤ 23 by O. Borodin and A. Ivanova. In this paper, we prove that every such polytope P satisfies h ( P ) ≤ 17 , which bound is sharp.

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Tri Lai Matt Blum conjectured that the- number of tilings of a hexagonal dungeon with side-lengths a , 2 a , b , a , 2 a , b (for b ≥ 2 a ) equals 1 3 2 a 2 1 4 ⌊ a 2 ∕ 2 ⌋ . Ciucu and the author of the present paper proved the conjecture by using Kuo’s graphical condensation method. In this paper, we investigate a 3-parameter refinement of the conjecture and its application to enumeration of tilings of several new types of the hexagonal dungeons.

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Martin Rolek, Zi-Xia Song A connected graph G with chromatic number t is double-critical if G ∖ { x , y } is ( t − 2 ) -colorable for each edge x y ∈ E ( G ) . The complete graphs are the only known examples of double-critical graphs. A long-standing conjecture of Erdős and Lovász from 1966, which is referred to as the Double-Critical Graph Conjecture, states that there are no other double-critical graphs. That is, if a graph G with chromatic number t is double-critical, then G is the complete graph on t vertices. This has been verified for t ≤ 5 , but remains open for t ≥ 6 . In this paper, we first prove that if G is a non-complete, double-critical graph with chromatic number t ≥ 6 , then no vertex of degree t + 1 is adjacent to a vertex of degree t + 1 , t + 2 , or t + 3 in G . We then use this result to show that the Double-Critical Graph Conjecture is true for double-critical graphs G with chromatic number t ≤ 8 if G is claw-free.

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Wojciech Wideł Graph G of order n is said to be pancyclic if it contains cycles of all lengths k for k ∈ { 3 , … , n } . A vertex v ∈ V ( G ) is called super-heavy if its degree in G is at least ( n + 1 ) ∕ 2 . For a given graph S we say that G is S - f 1 -heavy if for every induced subgraph K of G isomorphic to S and every two vertices u , v ∈ V ( K ) , d K ( u , v ) = 2 implies that at least one of them is super-heavy. For a family of graphs S we say that G is S - f 1 -heavy, if G is S - f 1 -heavy for every graph S ∈ S . Let H denote the hourglass, a graph consisting of two triangles that have exactly one vertex in common. In this paper we prove that every 2 -connected { K 1 , 3 , P 7 , H } - f 1 -heavy graph on at least nine vertices is pancyclic or missing only one cycle. This result extends the previous work by Faudree, Ryjáček and Schiermeyer.

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Hao Chen Inspired by Bondarenko’s counter-example to Borsuk’s conjecture, we notice some strongly regular graphs that provide examples of ball packings whose chromatic numbers are significantly higher than the dimensions. In particular, from generalized quadrangles we obtain unit ball packings in dimension q 3 − q 2 + q with chromatic number q 3 + 1 , where q is a prime power. This improves the previous lower bounds for the chromatic number of ball packings.

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): J. Rifà, V.A. Zinoviev We construct several classes of completely regular codes with different parameters, but identical intersection array. Given a prime power q and any two natural numbers a , b , we construct completely transitive codes over different fields with covering radius ρ = min { a , b } and identical intersection array, specifically, one code over F q r for each divisor r of a or b . As a corollary, for any prime power q , we show that distance regular bilinear forms graphs can be obtained as coset graphs from several completely regular codes with different parameters.

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Yiqiao Wang, Xiaoxue Hu, Weifan Wang The linear 2 -arboricity la 2 ( G ) of a graph G is the least integer k such that G can be partitioned into k edge-disjoint forests, whose components are paths of length at most 2. In this paper, we prove that every planar graph G with Δ = 10 has la 2 ( G ) ≤ 9 . Using this result, we correct an error in the proof of a result in Wang (2016), which says that every planar graph G satisfies la 2 ( G ) ≤ ⌈ ( Δ + 1 ) ∕ 2 ⌉ + 6 .

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Pierre Coupechoux In 1998, Karpovsky, Chakrabarty and Levitin introduced identifying codes to model fault diagnosis in multiprocessor systems (Karpovsky et al., 1998). In these codes, each vertex is identified by the vertices belonging to the code in its neighborhood. There exists a coloring variant as follows: a globally identifying coloring of a graph is a coloring such that each vertex is identified by the colors in its neighborhood. We aim at finding the maximum length of a cycle with such a coloring, given a fixed number of colors we can use. Parreau (2012) used Jackson’s work (Jackson, 1993) on universal cycles to give a lower bound of this length. In this article, we will adapt what Jackson did, to improve this result.

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Liliana Alcón, Marisa Gutierrez, Glenn Hurlbert Graph pebbling is a network model for transporting discrete resources that are consumed in transit. Deciding whether a given configuration on a particular graph can reach a specified target is NP-complete, even for diameter two graphs, and deciding whether the pebbling number has a prescribed upper bound is Π 2 P -complete. Recently we proved that the pebbling number of a split graph can be computed in polynomial time. This paper advances the program of finding other polynomial classes, moving away from the large tree width, small diameter case (such as split graphs) to small tree width, large diameter, continuing an investigation on the important subfamily of chordal graphs called k -trees. In particular, we provide a formula, that can be calculated in polynomial time, for the pebbling number of any semi-2-tree, falling shy of the result for the full class of 2-trees.

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Michał Dębski The strong chromatic index of a graph G , denoted by s ′ ( G ) , is the minimum possible number of colors in a coloring of the edges of G such that each color class is an induced matching. The corresponding fractional parameter is denoted by s f ′ ( G ) . For a bipartite graph G we have s f ′ ( G ) ≤ 1 . 5 Δ G 2 . This follows as an easy consequence of earlier results — the fractional variant of Reed’s conjecture and the theorem by Faudree, Gyárfás, Schelp and Tuza from 1990. Both these results are tight so it may seem that the bound 1 . 5 Δ G 2 is best possible. We break this “1.5 barrier”. We prove that s f ′ ( G ) ≤ 1 . 4762 Δ G 2 + Δ G 1 . 5 for every bipartite graph G . The main part of the proof is a structural lemma regarding cliques in L ( G ) 2 .

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Yun-Ping Deng, Yu-Qin Sun, Qiong Liu, Hai-Chao Wang Let S be a subset of finite cyclic group Z n not containing the identity element 0 with S = − S . Cayley graphs on Z n with respect to S are called circulant graphs and denoted by Cay ( Z n , S ) . In this paper, for connected non-complete circulant graphs Cay ( Z n , S ) of degree S = p − 1 with p prime, we give a necessary and sufficient condition for the existence of efficient dominating sets, and characterize all efficient dominating sets if exist. We also obtain similar results for Cay ( Z n , S ) of degree S = p q − 1 and p m − 1 , where p , q are primes, m is a positive integer, and S + 1 is relatively prime to n S + 1 . Moreover, we give a necessary and sufficient condition for the existence of efficient dominating sets in Cay ( Z n , S ) of order n = p k q , p 2 q 2 , p q r , p 2 q r , p q r s and degree S , where p , q , r , s are distinct primes, k is a positive integer, and S + 1 is relatively prime to n S + 1 .

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Esmeralda Năstase, Papa Sissokho Let n and t be positive integers with t < n , and let q be a prime power. A partial ( t − 1 ) -spread of PG ( n − 1 , q ) is a set of ( t − 1 ) -dimensional subspaces of PG ( n − 1 , q ) that are pairwise disjoint. Let r ≡ n ( mod t ) with 0 ≤ r < t , and let Θ i = ( q i − 1 ) ∕ ( q − 1 ) . We essentially prove that if 2 ≤ r < t ≤ Θ r , then the maximum size of a partial ( t − 1 ) -spread of PG ( n − 1 , q ) is bounded from above by ( Θ n − Θ t + r ) ∕ Θ t + q r − ( q − 1 ) ( t − 3 ) + 1 . We actually give tighter bounds when certain divisibility conditions are satisfied. These bounds improve on the previously known upper bound for the maximum size partial ( t − 1 )-spreads of PG ( n − 1 , q ) ; for instance, when ⌈ Θ r 2 ⌉ + 4 ≤ t ≤ Θ r and q > 2 . The exact value of the maximum size partial ( t − 1 ) -spread has been recently determined for t > Θ r by the authors of this paper (see Năstase and Sissokho [21] (2016)).

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Huilan Chang, Wei-Cheng Lan To identify splice sites in a genome, Cicalese et al. (2005) studied interval group testing where all items in the search space are linearly ordered and each of them is either positive or negative. The goal is to identify all positive ones by interval group tests, each asking a query of the type “does a set of consecutive items contain any positive one?” We complete the study of error-tolerant one- and two-stage approaches by dealing with some cases which have not been well studied. Motivated by applications to DNA sequencing, group testing for consecutive positives has been proposed by Balding and Torney (1997) and Colbourn (1999) where n items are linearly ordered and up to p positive items are consecutive in the order. Juan and Chang (2008) provided an optimal sequential algorithm that consists of interval group tests. In this paper, we study error-tolerant one- and two-stage interval group testing for consecutive positives. Based on some results in the literature, we derive optimal nonadaptive algorithms. For two-stage approach, the number of interval group tests used is asymptotically upper bounded by e n which differs by a factor of e from the lower bound, where e is the number of errors allowed.

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Yucong Tang, Guiying Yan A Hamilton ℓ -cycle in a k -uniform hypergraph of n -vertex is an ordering of all vertices, combined with an ordered subset C of edges, such that any two consecutive edges share exactly ℓ vertices and each edge in C contains k consecutive vertices. A classic result of O. Ore in 1960 is that if the degree sum of any two independent vertices in an n -vertex graph is at least n , then the graph contains a Hamiltonian cycle. Naturally, we consider to generalize it to hypergraphs situation. In this paper, we prove the following theorems. (i) For any n ≥ 4 k 2 , there is an n -vertex k -uniform hypergraph, with degree sum of any two strongly independent sets of k − 1 vertices bigger than 2 n − 4 ( k − 1 ) , contains no Hamilton l -cycle, 1 ≤ ℓ ≤ k − 1 . (ii) If the degree sum of two weakly independent sets of k − 1 vertices in an n -vertex k -uniform hypergraph is ( 1 + o ( 1 ) ) n , then the hypergraph contains a Hamilton ( k − 1 ) -cycle, where two distinct sets of k − 1 vertices are weakly (strongly) independent if there exist no edge containing the union of them (intersecting both of them).

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Michael Gentner, Dieter Rautenbach Let G be a graph with degree sequence d 1 ≥ ⋯ ≥ d n . Slater proposed s ℓ ( G ) = min { s : ( d 1 + 1 ) + ⋯ + ( d s + 1 ) ≥ n } as a lower bound on the domination number γ ( G ) of G . We show that deciding the equality of γ ( G ) and s ℓ ( G ) for a given graph G is NP-complete but that one can decide efficiently whether γ ( G ) > s ℓ ( G ) or γ ( G ) ≤ ln n ( G ) s ℓ ( G ) + 1 s ℓ ( G ) . For real numbers α and β with α ≥ max { 0 , β } , let G ( α , β ) be the class of non-null graphs G such that every non-null subgraph H of G has at most α n ( H ) − β many edges. Generalizing a result of Desormeaux, Haynes, and Henning, we show that γ ( G ) ≤ ( 2 α + 1 ) s ℓ ( G ) − 2 β for every graph G in G ( α , β ) with α ≤ 3 2 . Furthermore, we show that γ ( G ) ∕ s ℓ ( G ) is bounded for graphs G in G ( α , β ) if and only if α < 2 . For an outerplanar graph G with PubDate: 2017-03-25T08:00:20Z

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Jason Altschuler, Elizabeth Yang In 1984, Kelly and Oxley introduced the model of a random representable matroid M [ A n ] corresponding to a random matrix A n ∈ F q m ( n ) × n , whose entries are drawn independently and uniformly from F q . Whereas properties such as rank, connectivity, and circuit size have been well-studied, forbidden minors have not yet been analyzed. Here, we investigate the asymptotic probability as n → ∞ that a fixed F q -representable matroid M is a minor of M [ A n ] . (We always assume m ( n ) ≥ rank ( M ) for all sufficiently large n , otherwise M can never be a minor of the corresponding M [ A n ] .) When M is free, we show that M is asymptotically almost surely (a.a.s.) a minor of M [ A n ] . When M is not free, we show a phase transition: M is a.a.s. a minor if n − m ( n ) → ∞ , but is a.a.s. not if m ( n ) − n → ∞ . In the more general settings of m ≤ n and m > n , we give lower and upper bounds, respectively, on both the asymptotic and non-asymptotic probabilities that M is a minor of M [ A n ] . The tools we develop to analyze matroid operations and minors of random matroids may be of independent interest. Our results directly imply that M [ A n ] is a.a.s. not contained in any proper, minor-closed class M of F q -representable matroids, provided: (i) n − m ( n ) → ∞ , and (ii) m ( n ) is at least the minimum rank of any F q -representable forbidden minor of PubDate: 2017-03-25T08:00:20Z

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Olivier Baudon, Jakub Przybyło, Mohammed Senhaji, Elżbieta Sidorowicz, Éric Sopena, Mariusz Woźniak Let γ : E ( G ) ⟶ N ∗ = N ∖ { 0 } be an edge colouring of a graph G and σ γ : V ( G ) ⟶ N ∗ the vertex colouring given by σ γ ( v ) = ∑ e ∋ v γ ( e ) for every v ∈ V ( G ) . A neighbour-sum-distinguishing edge-colouring of G is an edge colouring γ such that for every edge u v in G , σ γ ( u ) ≠ σ γ ( v ) . The neighbour-sum-distinguishing edge-colouring game on G is the 2-player game defined as follows. The two players, Alice and Bob, alternately colour an uncoloured edge of G . Alice wins the game if, when all edges are coloured, the so-obtained edge colouring is a neighbour-sum-distinguishing edge-colouring of G . Otherwise, Bob wins. In this paper we study the neighbour-sum-distinguishing edge-colouring game on various classes of graphs. In particular, we prove that Bob wins the game on the complete graph K n , n ≥ 3 , whoever starts the game, except when n = 4 . In that case, Bob wins the game on K 4 if and only if Alice starts the game.

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Rennan Dantas, Frédéric Havet, Rudini M. Sampaio A set C ⊂ V ( G ) is an identifying code in a graph G if for all v ∈ V ( G ) , C [ v ] ≠ ∅ , and for all distinct u , v ∈ V ( G ) , C [ u ] ≠ C [ v ] , where C [ v ] = N [ v ] ∩ C and N [ v ] denotes the closed neighborhood of v in G . The minimum density of an identifying code in G is denoted by d ∗ ( G ) . Given a positive integer k , let T k be the triangular grid with k rows. In this paper, we prove that d ∗ ( T 1 ) = d ∗ ( T 2 ) = 1 ∕ 2 , d ∗ ( T 3 ) = d ∗ ( T 4 ) = 1 ∕ 3 , d ∗ ( T 5 ) = 3 ∕ 10 , d ∗ ( T 6 ) = 1 ∕ 3 and d ∗ ( T k ) = 1 ∕ 4 + 1 ∕ ( 4 k ) for every k ≥ 7 odd. Moreover, we prove that 1 ∕ 4 + 1 ∕ ( 4 k ) ≤ d ∗ ( T k ) ≤ 1 ∕ 4 + 1 ∕ ( 2 k ) for every k ≥ 8 even. We conjecture that d ∗ ( T k )... PubDate: 2017-03-25T08:00:20Z

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Ana Paulina Figueroa, César Hernández-Cruz, Mika Olsen A minimum feedback arc set of a digraph D is a minimum set of arcs which removal leaves the resultant graph free of directed cycles; its cardinality is denoted by τ 1 ( D ) . The acyclic disconnection of D , ω ⃗ ( D ) , is defined as the maximum number of colors in a vertex coloring of D such that every directed cycle of D contains at least one monochromatic arc. In this article we study the relationship between the minimum feedback arc set and the acyclic disconnection of a digraph, we prove that the acyclic disconnection problem is NP -complete. We define the acyclic disconnection and the minimum feedback for graphs. We also prove that ω ⃗ ( G ) + τ 1 ( G ) = V ( G ) if G is a wheel, a grid or an outerplanar graph.

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Rongquan Feng, He Huang, Sanming Zhou A perfect code in a graph Γ = ( V , E ) is a subset C of V that is an independent set such that every vertex in V ∖ C is adjacent to exactly one vertex in C . A total perfect code in Γ is a subset C of V such that every vertex of V is adjacent to exactly one vertex in C . A perfect code in the Hamming graph H ( n , q ) agrees with a q -ary perfect 1-code of length n in the classical setting. In this paper we give a necessary and sufficient condition for a circulant graph of degree p − 1 to admit a perfect code, where p is an odd prime. We also obtain a necessary and sufficient condition for a circulant graph of order n and degree p l − 1 to have a perfect code, where p is a prime and p l the largest power of p dividing n . Similar results for total perfect codes are also obtained in the paper.

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Gabriel Nivasch Let L be a set of n lines in the plane, and let C be a convex curve in the plane, like a circle or a parabola. The zone of C in L , denoted Z ( C , L ) , is defined as the set of all cells in the arrangement A ( L ) that are intersected by C . Edelsbrunner et al. (1992) showed that the complexity (total number of edges or vertices) of Z ( C , L ) is at most O ( n α ( n ) ) , where α is the inverse Ackermann function. They did this by translating the sequence of edges of Z ( C , L ) into a sequence S that avoids the subsequence a b a b a . Whether the worst-case complexity of Z ( C , L ) is only linear is a longstanding open problem. Since the relaxation of the problem to pseudolines does have a Θ ( n α ( n ) ) bound, any proof of O ( n ) for the case of straight lines must necessarily use geometric arguments. In this paper we present some such geometric arguments. We show that, if C is a circle, then certain configurations of straight-line segments with endpoints on C are impossible. In particular, we show that there exists a Hart–Sharir sequence that cannot appear as a subsequence of S . The Hart–Sharir sequences are essentially the only known way to construct a b a b a -free sequences of superlinear length. Hence, if it could be shown that every family of a b a b a -free sequences of superlinear-length eventually contains all Hart–Sharir sequences, it would follow that the complexity of Z ( C , L ) is O ( n ) whenever C is a circle.

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Gregory J. Puleo We prove that if M is a maximal k -edge-colorable subgraph of a multigraph G and if F = { v ∈ V ( G ) : d M ( v ) ≤ k − μ ( v ) } , then d F ( v ) ≤ d M ( v ) for all v ∈ V ( G ) with d M ( v ) < k . (When G is a simple graph, the set F is just the set of vertices having degree less than k in M .) This implies Vizing’s Theorem as well as a special case of Tuza’s Conjecture on packing and covering of triangles. A more detailed version of our result also implies Vizing’s Adjacency Lemma for simple graphs.

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Yue Zhou For any cyclic group G , we show the nonexistence of distinct subsets A , B and C ⊆ G such that G has the group factorizations G = A B = A C = B C . As a consequence, there are exactly two sequences in an arbitrary perfect asynchronous channel hopping system.

Abstract: Publication date: July 2017 Source:Discrete Mathematics, Volume 340, Issue 7 Author(s): Anthony D. Forbes, Terry S. Griggs, Tamsin J. Forbes We examine the design spectra for the vertex–edge graphs of some Archimedean solids. In particular, we complete the computation of the spectrum for the truncated cuboctahedron (1 or 64 modulo 144). We extend the known spectrum of the rhombicosidodecahedron by showing that there exist designs of order 81 modulo 240. We add residue class 81 modulo 120 to the known spectra of the icosidodecahedron and the snub cube, each with one possible exception. We add residue class 145 modulo 180 to the known spectra of the truncated dodecahedron and the truncated icosahedron, each with two possible exceptions. Finally, we exhibit the first explicit examples of snub dodecahedron designs.

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Sándor Z. Kiss, Csaba Sándor For a set of nonnegative integers S let R S ( n ) denote the number of unordered representations of the integer n as the sum of two different terms from S . In this paper we focus on partitions of the natural numbers into two sets affording identical representation functions. We solve a recent problem of Chen and Lev.

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Marcel Abas, Tomáš Vetrík Let C d , k be the largest number of vertices in a Cayley digraph of degree d and diameter k , and let B C d , k be the largest order of a bipartite Cayley digraph for given d and k . For every degree d ≥ 2 and for every odd k we construct Cayley digraphs of order 2 k ⌊ d 2 ⌋ k and diameter at most k , where k ≥ 3 , and bipartite Cayley digraphs of order 2 ( k − 1 ) ⌊ d 2 ⌋ k − 1 and diameter at most k , where k ≥ 5 . These constructions yield the bounds C d , k ≥ 2 k ⌊ d 2 ⌋ k for odd k ≥ 3 and d ≥ 3 k 2 k + 1 , and B C d , k ≥ 2 ( k − 1 ) ⌊ d 2 ⌋ k − 1 for odd k ≥ 5 and d ≥ 3 k − 1 k − 1 + 1 . Our constructions give the best currently known bounds on the orders of large Cayley digraphs and bipartite Cayley digraphs of given degree and odd diameter k ≥ 5 . In our proofs we use new techniques based on properties of group automorphisms of direct products of abelian groups.

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Oliver Cooley, Richard Mycroft We prove that any 3-uniform hypergraph whose minimum vertex degree is at least 5 9 + o ( 1 ) n 2 admits an almost-spanning tight cycle, that is, a tight cycle leaving o ( n ) vertices uncovered. The bound on the vertex degree is asymptotically best possible. Our proof uses the hypergraph regularity method, and in particular a recent version of the hypergraph regularity lemma proved by Allen, Böttcher, Cooley and Mycroft.

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Sherry H.F. Yan, Guizhi Qin, Zemin Jin, Robin D.P. Zhou In this paper, we are mainly concerned with the enumeration of ( 2 k + 1 , 2 k + 3 ) -core partitions with distinct parts. We derive the number and the largest size of such partitions, confirming two conjectures posed by Straub.

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Zhihui Jiao, Hong Wang, Jin Yan In this paper, we prove that for any positive integer k and for any simple graph G of order at least 3 k , if d ( x ) + d ( y ) ≥ 4 k for every pair of vertices x and y of distance 2 in G , then with one exception, G contains k disjoint cycles. This generalizes a former result of Corrádi and Hajnal (1963).

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Yusuke Suzuki We discuss the existence of minors of given graphs in optimal 1 -planar graphs. As our first main result, we prove that for any graph H , there exists an optimal 1 -planar graph which contains H as a topological minor. Next, we consider minors of complete graphs. It is easily obtained from Mader’s result (Mader, 1968) that every optimal 1 -planar graph has a K 6 -minor. In the paper, we characterize optimal 1 -planar graphs having no K 7 -minor.

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Raúl M. Falcón, Rebecca J. Stones An r × s partial Latin rectangle ( l i j ) is an r × s matrix containing elements of { 1 , 2 , … , n } ∪ { ⋅ } such that each row and each column contain at most one copy of any symbol in { 1 , 2 , … , n } . An entry is a triple ( i , j , l i j ) with l i j ≠ ⋅ . Partial Latin rectangles are operated on by permuting the rows, columns, and symbols, and by uniformly permuting the coordinates of the set of entries. The stabilizers under these operations are called the autotopism group and the autoparatopism group, respectively. We develop the theory of symmetries of partial Latin rectangles, introducing the concept of a partial Latin rectangle graph. We give constructions of m -entry partial Latin rectangles with trivial autotopism groups for all possible autoparatopism groups (up to isomorphism) when: (a) r = s = n , i.e., partial Latin squares, (b) r = 2 and s = n , and (c) r = 2 and s ≠ n .

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Nicholas Harvey, Christopher Liaw The Lovász Local Lemma is a powerful probabilistic tool used to prove the existence of combinatorial structures which avoid a set of constraints. A standard way to apply the local lemma is to prove that the set of constraints satisfy a lopsidependency condition and obtain a lopsidependency graph. For instance, Erdős and Spencer used this framework to posit the existence of Latin transversals in matrices provided no symbol appears too often in the matrix. The local lemma has been used in various ways to infer the existence of rainbow Hamilton cycles in complete graphs when each colour is used at most O ( n ) times. However, the existence of a lopsidependency graph for Hamilton cycles has neither been proved nor refuted. All previous approaches have had to prove a variant of the local lemma or reduce the problem of finding Hamilton cycles to finding another combinatorial structure, such as Latin transversals. In this paper, we revisit the question of whether or not Hamilton cycles have a lopsidependency graph and give a positive answer for this question. We also use the resampling oracle framework of Harvey and Vondrák to give a polynomial time algorithm for finding rainbow Hamilton cycles in complete graphs.

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Jason Brown, Lucas Mol Given a graph G in which each edge fails independently with probability q ∈ [ 0 , 1 ] , the all-terminal reliability of G is the probability that all vertices of G can communicate with one another, that is, the probability that the operational edges span the graph. The all-terminal reliability is a polynomial in q whose roots (all-terminal reliability roots) were conjectured to have modulus at most 1 by Brown and Colbourn. Royle and Sokal proved the conjecture false, finding roots of modulus larger than 1 by a slim margin. Here, we present the first nontrivial upper bound on the modulus of any all-terminal reliability root, in terms of the number of vertices of the graph. We also find all-terminal reliability roots of larger modulus than any previously known. Finally, we consider the all-terminal reliability roots of simple graphs; we present the smallest known simple graph with all-terminal reliability roots of modulus greater than 1 , and we find simple graphs with all-terminal reliability roots of modulus greater than 1 that have higher edge connectivity than any previously known examples.

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Klaus Metsch The notion of Cameron–Liebler line classes was generalized in Rodgers et al. (0000) to Cameron–Liebler k -classes, where k = 1 corresponds to the line classes. Such a set consists of x 2 k + 1 k q subspaces of dimension k in PG ( 2 k + 1 , q ) where k ≥ 1 and x ≥ 0 are integers such that every regular k -spread of PG ( 2 k + 1 , q ) contains exactly x subspaces from the set. Examples are known for x ≤ 2 . The authors of Rodgers et al. (0000) show that there are no Cameron–Liebler k -classes when k = 2 and 3 ≤ x ≤ q , or when 3 ≤ k ≤ q log q − q and 3 ≤ x ≤ q ∕ 2 3 . We improve these results by weakening the condition on the upper bound for x to a bound that is linear in q . For this, we use a technique that was originally used to extend nets to affine planes.

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Hwa-Young Lee, Bokhee Im, Jonathan D.H. Smith Triply stochastic cubic tensors, or sharply transitive sets of doubly stochastic matrices, are decompositions of the all-ones matrix as the sum of an ordered set of bistochastic matrices. They combine to yield so-called weak approximate quasigroups and Latin squares. Approximate symmetry is implemented by the stochastic matrix actions of quasigroups on homogeneous spaces, thereby extending the concept of exact symmetry as implemented by permutation matrix actions of groups on coset spaces. Now approximate quasigroups and Latin squares are described as being strong if they occur within quasigroup actions. We study these weak and strong objects, in particular examining the location of the latter within the polytope of triply stochastic cubic tensors. We also establish the rudiments of an algebraic structure theory for approximate quasigroups. Upon relaxation from probability distributions to their supports, approximate quasigroups furnish non-associative analogues of (set-theoretical) hypergroups.

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Nicholas J. Cavenagh, Terry S. Griggs We consider the problem of classifying trades in Steiner triple systems such that each block of the trade contains one of three fixed elements. We show that the fundamental building blocks for such trades are 3 -regular graphs that are 1 -factorisable. In the process we also generate all possible 2 - and 3 -way simultaneous edge colourings of graphs with maximum degree 3 using at most 3 colours, where multiple edges but not loops are allowed. Moreover, we generate all possible Latin trades within three rows.

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Weihua He, Weihua Yang The line graph is a very popular research object in graph theory, in complex networks and also in social networks recently. In this paper, we show that if a line graph is Hamiltonian-connected, then the graphs in a special family of spanning subgraphs of the line graph are still Hamiltonian-connected. As an important corollary we prove that there exist at least m a x { 1 , ⌊ 1 8 δ ( G ) ⌋ − 1 } edge-disjoint Hamiltonian paths between any two vertices in a Hamiltonian-connected line graph L ( G ) .

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Junye Ma, Bicheng Zhang, Jinwang Liu, Jutao Zhao In this paper, we study 24 possibilities of directed strongly regular graphs with adjacency matrix of rank 6, our proof is based on two effective algorithms. At last, we get 4 families of directed strongly regular graphs with realizable parameter sets, exclude 20 families of directed strongly regular graphs with feasible parameter sets; partially solve the classification problem of directed strongly regular graphs with adjacency matrix of rank 6.

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Giorgio Donati, Nicola Durante Let A and B be two points of PG ( 3 , q n ) and let Φ be a collineation between the stars of lines with vertices A and B that fixes the line A B . In this paper we prove that the set C of points of intersection of corresponding lines under Φ , under the hypothesis that C generates PG ( 3 , q n ) , is either the union of two degenerate C F m -sets in two planes through the line A B (see Donati and Durante, 2014) or the union of a scattered GF ( q ) -linear set of rank n + 2 with the line A B . We also determine the intersection configurations of two scattered GF ( q ) -linear sets of rank n + 2 of PG ( 3 , q n ) both meeting the line A B in a GF ( q ) -linear set of pseudoregulus type with transversal points A and B .

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Taylor Ball, Robert W. Bell, Jonathan Guzman, Madeleine Hanson-Colvin, Nikolas Schonsheck We prove that the cop number of every generalized Petersen graph is at most 4. The strategy is to play a modified game of cops and robbers on an infinite cyclic covering space where the objective is to capture the robber or prevent the robber from visiting any vertex infinitely often. We also prove that finite isometric subtrees of any graph are 1-guardable, and we apply this to determine the exact cop number of some families of generalized Petersen graphs. Additionally, we extend these ideas to prove that the cop number of any connected I -graph is at most 5.

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Lili Mu, Yi Wang Let ( a 0 , a 1 , a 2 , … ) be the sequence of Catalan-like numbers. We evaluate the Hankel determinants of the shifted sequence ( 0 , a 0 , a 1 , a 2 , … ) . As an application, we settle Barry’s three conjectures about Hankel determinant evaluations of certain sequences in a unified approach. We also provide some Somos sequences by means of Hankel determinants of the (shifted) Catalan-like numbers.

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): John A. Arredondo, Camilo Ramírez Maluendas, Ferrán Valdez We prove that infinite regular and chiral maps can only exist on surfaces with one end. Moreover, we prove that an infinite regular or chiral map on an orientable surface with positive genus, can only be realized on the Loch Ness monster, that is, the topological surface of infinite genus with one end.

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Jürgen Bierbrauer, S. Marcugini, F. Pambianco A ternary [ 66 , 10 , 36 ] 3 -code admitting the Mathieu group M 12 as a group of automorphisms has recently been constructed by N. Pace, see Pace (2014). We give a construction of the Pace code in terms of M 12 as well as a combinatorial description in terms of the small Witt design, the Steiner system S ( 5 , 6 , 12 ) . We also present a proof that the Pace code does indeed have minimum distance 36 .

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Andrea Munaro Line graphs constitute a rich and well-studied class of graphs. In this paper, we focus on three different topics related to line graphs of subcubic triangle-free graphs. First, we show that any such graph G has an independent set of size at least 3 V ( G ) ∕ 10 , the bound being sharp. As an immediate consequence, we have that any subcubic triangle-free graph G , with n i vertices of degree i , has a matching of size at least 3 n 1 ∕ 20 + 3 n 2 ∕ 10 + 9 n 3 ∕ 20 . Then we provide several approximate min-max theorems relating cycle-transversals and cycle-packings of line graphs of subcubic triangle-free graphs. This enables us to prove Jones’ Conjecture for claw-free graphs with maximum degree 4 . Finally, we concentrate on the computational complexity of Feedback Vertex Set, Hamiltonian Cycle and Hamiltonian Path for subclasses of line graphs of subcubic triangle-free graphs.

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Ruonan Li, Hajo Broersma, Chuandong Xu, Shenggui Zhang Let G be an edge-colored graph. The minimum color degree of G is the minimum number of different colors appearing on the edges incident with the vertices of G . In this paper, we study the existence of properly edge-colored cycles in (not necessarily properly) edge-colored complete graphs. Fujita and Magnant (2011) conjectured that in an edge-colored complete graph on n vertices with minimum color degree at least ( n + 1 ) ∕ 2 , each vertex is contained in a properly edge-colored cycle of length k , for all k with 3 ≤ k ≤ n . They confirmed the conjecture for k = 3 and k = 4 , and they showed that each vertex is contained in a properly edge-colored cycle of length at least 5 when n ≥ 13 , but even the existence of properly edge-colored Hamilton cycles is unknown (in complete graphs that satisfy the conditions of the conjecture). We prove a cycle extension result that implies that each vertex is contained in a properly edge-colored cycle of length at least the minimum color degree.

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Beifang Chen, Jue Wang, Thomas Zaslavsky It is well known that each nonnegative integral flow on a graph can be decomposed into a sum of nonnegative graphic circuit flows, which cannot be further decomposed into nonnegative integral sub-flows. This is equivalent to saying that the indecomposable flows on graphs are those graphic circuit flows. Turning from graphs to signed graphs, the indecomposable flows are much richer than those of unsigned graphs. This paper gives a complete description of indecomposable flows on signed graphs from the viewpoint of resolution of singularities by means of double covering graph.

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Byung-Geun Oh For a connected simple graph embedded into a 2-sphere, we show that the number of vertices of the graph is less than or equal to 380 if the degree of each vertex is at least three, the combinatorial vertex curvature is positive everywhere, and the graph is different from prisms and antiprisms. This gives a new upper bound for the constant brought up by DeVos and Mohar in their paper from 2007. We also show that if a graph is embedded into a projective plane instead of a 2-sphere but satisfies the other properties listed above, then the number of vertices is at most 190.

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Tricia Muldoon Brown We describe a class of fixed polyominoes called k -omino towers that are created by stacking rectangular blocks of size k × 1 on a convex base composed of these same k -omino blocks. By applying a partition to the set of k -omino towers of fixed area k n , we give a recurrence on the k -omino towers therefore showing the set of k -omino towers is enumerated by a Gauss hypergeometric function. The proof in this case implies a more general hypergeometric identity with parameters similar to those given in a classical result of Kummer.

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Eun Ju Cheon, Masaaki Homma, Seon Jeong Kim, Namyong Lee In Homma and Kim (2010), an upper bound of the number of rational points on a plane curve of degree d over F q is found. Some examples attaining the bound are given in Homma and Kim (2010), whose degrees are q + 2 , q + 1 , q , q − 1 , q − 1 (when q is a square), and 2 . In this paper, we consider an actual upper bound on such numbers for curves of low degree for q ≤ 7 . Also we give explicit examples of curves attaining the sharp bound for each d and q .

Abstract: Publication date: June 2017 Source:Discrete Mathematics, Volume 340, Issue 6 Author(s): Rémi Desgranges, Kolja Knauer In this note we determine the set of expansions such that a partial cube is planar if and only if it arises by a sequence of such expansions from a single vertex. This corrects a result of Peterin.