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Abstract: Abstract Let G be a finite group. A number of graphs with the vertex set G have been studied, including the power graph, enhanced power graph, and commuting graph. These graphs form a hierarchy under the inclusion of edge sets, and it is useful to study them together. In addition, several authors have considered modifying the definition of these graphs by choosing a natural equivalence relation on the group such as equality, conjugacy, or equal orders, and joining two elements if there are elements in their equivalence class that are adjacent in the original graph. In this way, we enlarge the hierarchy into a second dimension. Using the three graph types and three equivalence relations mentioned gives nine graphs, of which in general only two coincide; we find conditions on the group for some other pairs to be equal. These often define interesting classes of groups, such as EPPO groups, 2-Engel groups, and Dedekind groups. We study some properties of graphs in this new hierarchy. In particular, we characterize the groups for which the graphs are complete, and in most cases, we characterize the dominant vertices (those joined to all others). Also, we give some results about universality, perfectness, and clique number. PubDate: 2022-05-23

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Abstract: Abstract Let \(N=2^\alpha 3^\beta\) . The normalizer \(\Gamma _B(N)\) of \(\Gamma _0(N)\) in \(PSL(2,\mathbb {R})\) is the triangle group \((2,4,\infty )\) for \(\alpha =1,3,5,7\) ; \(\beta =0,2\) and the triangle group \((2,6,\infty )\) for \(\alpha =0,2,4,6\) ; \(\beta =1,3\) . In this paper we examine relationship between the normalizer and the regular maps. We define a family of subgroups of the normalizer and then we study maps with quadrilateral and hexagonal faces using these subgroups and calculating the associated arithmetic structure. PubDate: 2022-05-21

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Abstract: Abstract Let \(A_n=(a_1,a_2,\ldots ,a_n)\) and \(B_n=(b_1,b_2,\ldots ,b_n)\) be nonnegative integer sequences with \(A_n\le B_n\) . The purpose of this note is to give a good characterization such that every integer sequence \(\pi =(d_1,d_2,\ldots d_n)\) with even sum and \(A_n\le \pi \le B_n\) is graphic. This solves a forcible version of problem posed by Niessen and generalizes the Erdős–Gallai theorem. PubDate: 2022-05-19

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Abstract: Abstract For each \(n\ge 14\) , we provide an example of a linklessly embeddable, Tutte-4-connected graph of order n. We start with a linklessly embeddable, Tutte-4-connected graph of order fourteen, and we perform 4-vertex splittings to inductively build the family of triangle free, 4-connected graphs. We prove the graphs we constructed are linklessly embeddable, as minors of clique sums over \(K_4\) of linklessly embeddable graphs. PubDate: 2022-05-19

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Abstract: Abstract Let \(D=(V,E)\) and \(H=(U,F)\) be digraphs and consider a colouring of the arcs of D with the vertices of H; we say that D is H coloured. We study a natural generalisation of the notion of kernel, as introduced by V. Neumann and Morgenstern (1944), to prove that If every cycle of D is an H-cycle, then D has an H-kernel by walks. As a consequence of this, we are able to give several sufficient conditions for the existence of H-kernels by walks; in particular, we solve partially a conjecture by Bai et al. in this context [2]; viz., they work with complete H without loops, and use paths rather than walks, so whenever the existence of H-paths is implied by the existence of H-walks our result can be use to corroborate Bai’s conjecture—in particular, if D is two coloured, and each cycle is alternating, then each alternating walk contains an alternating path. PubDate: 2022-05-16

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Abstract: Abstract A labelling of a graph G is a mapping \(\pi :S \rightarrow {\mathcal {L}}\) , where \({\mathcal {L}}\subset {\mathbb {R}}\) and \(S\in \{E(G), V(G)\cup E(G)\}\) . If \(S=E(G)\) , \(\pi\) is an \({\mathcal {L}}\) -edge-labelling and, if \(S=V(G)\cup E(G)\) , \(\pi\) is an \({\mathcal {L}}\) -total-labelling. For each \(v\in V(G)\) , the colour of v under \(\pi\) is defined as \(c_{\pi }(v) = \sum _{uv \in E(G)}{\pi (uv)}\) if \(\pi\) is an \({\mathcal {L}}\) -edge-labelling; and \(c_{\pi }(v) = \pi (v)+\sum _{uv \in E(G)}{\pi (uv)}\) if \(\pi\) is an \({\mathcal {L}}\) -total-labelling. The pair \((\pi ,c_{\pi })\) is a neighbour-distinguishing \({\mathcal {L}}\) -edge-labelling (neighbour-distinguishing \({\mathcal {L}}\) -total-labelling) if \(\pi\) is an \({\mathcal {L}}\) -edge-labelling ( \({\mathcal {L}}\) -total-labelling) and \(c_{\pi }(u)\ne c_{\pi }(v)\) for every edge \(uv \in E(G)\) . In this work, we show that split graphs, regular cobipartite graphs, complete multipartite graphs and cubic graphs have neighbour-distinguishing \(\{a,b,c\}\) -edge-labellings, for distinct \(a,b,c \in {\mathbb {R}}\) (in some cases \(a,b,c \ge 0\) ). For split graphs and regular cobipartite graphs we also prove they admit neighbour-distinguishing \(\{a,b\}\) -total-labellings. Furthermore, we show that flower snarks and some subfamilies of split graphs and regular cobipartite graphs have neighbour-distinguishing \(\{a,b\}\) -edge-labellings and prove that some families of split graphs do not have neighbour-distinguishing PubDate: 2022-05-16

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Abstract: Abstract The neighborhood complex \(\mathcal {N}(G)\) of a graph G was introduced by L. Lovász in his proof of Kneser conjecture. He proved that for any graph G, 2 $$\begin{aligned} \chi (G) \ge conn(\mathcal {N}(G))+3. \end{aligned}$$ In this article we show that for a class of exponential graphs the bound given in (2) is tight. Further, we show that the neighborhood complexes of these exponential graphs are spheres up to homotopy. We were also able to find a class of exponential graphs, which are homotopy test graphs. In 1966, Hedetniemi conjectured that the chromatic number of the categori-cal product of two graphs is the minimum of the chromatic number of the factors. In 2019, Shitov [26] gave a counterexample to this conjecture. Let M(G) denotes the Mycielskian of a graph G. We show that, for any graph G containing \(M(M(K_n))\) as a subgraph and for any graph H, if \(\chi (G \times H) = n+1\) , then \(\min \{\chi (G), \chi (H)\} = n+1\) . Therefore, we enrich the family of graphs satisfying the Hedetniemi’s conjecture. PubDate: 2022-05-13

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Abstract: Abstract This paper studies an optimization problem of the fashion game on graphs on surfaces, especially on planar graphs. There are rebel players in a graph G. All players choose their actions from an identical set of the two symmetric actions, say \(\{0,1\}\) . An action profile of G is a mapping \(\pi :V(G)\rightarrow \{0,1\}\) . A rebel likes people having the different action with her and dislikes people having the same action. The utility \(u(v,\pi )\) of a player v under the action profile \(\pi\) is the number of neighbors she likes minus the number of neighbors she dislikes. Let \(\phi :V(G)\rightarrow {\mathbb {Z}}\) be a function. The \(\phi\) -satisfiability problem is to determine whether a graph has an action profile under which each player v has a utility at least \(\phi (v)\) . Let t be an integer. The t-satisfiability problem is the specialized \(\phi\) -satisfiability problem when \(\phi (v)=t\) , for each \(v\in V(G)\) . The utility of G, denoted by u(G), is defined to be the maximum t such that G is t-satisfiable. Let \(\eta : V(G)\rightarrow {\mathbb {N}}\) be a function. A mapping \(c:\ V(G)\rightarrow \{0,1\}\) is called an \(\eta\) -defective 2-coloring of G if every \(v\in V(G)\) has at most \(\eta (v)\) neighbors that have the same color with it. For graphs embeddable in surfaces, upper bounds of their utilities are given. The graphs embeddable in the torus or the Klein bottle whose utilities reach their upper bounds are determined. The t-satisfiability problem for graphs embeddable in the plane, the projective plane, the torus, or the Klein bottle is NP-complete if \(t\in \{1,2,3\}\) , and is polynomial time solvable otherwise. We design a dynamic programming algorithm that solves the \(\phi\) -satisfiability problem for outerplanar graphs in \(O( V(G) ^3)\) time. This algorithm can also solve the \(\eta\) -defective 2-coloring problem for outerplanar graphs. PubDate: 2022-05-12

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Abstract: Abstract We compute the equivariant Euler characteristics of the buildings for the symplectic groups over finite fields. PubDate: 2022-05-11

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Abstract: Abstract A polynomial is said to be unimodal if its coefficients are non-decreasing and then non-increasing. The domination polynomial of a graph G is the generating function of the number of dominating sets of each cardinality in G, and its coefficients have been conjectured to be unimodal. In this paper we will show the domination polynomial of paths, cycles and complete multipartite graphs are unimodal, and that the domination polynomial of almost every graph is unimodal with mode \( \lceil \frac{n}{2}\rceil \) . PubDate: 2022-05-09

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Abstract: Abstract In this paper, we classify all commutative weakly distance-regular digraphs of girth g and one type of arcs under the assumption that \(p_{(1,g-1),(1,g-1)}^{(2,g-2)}\ge k_{1,g-1}-2\) . In consequence, Yang et al. (J Comb Theory Ser A 160:288–315, 2018, Theorem 1.1) is partially generalized by our result. PubDate: 2022-05-04

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Abstract: Abstract The 1-2-3 Conjecture asks whether almost all graphs can be (edge-)labelled with 1, 2, 3 so that no two adjacent vertices are incident to the same sum of labels. In the last decades, several aspects of this problem have been studied in literature, including more general versions and slight variations. Notable such variations include the List 1-2-3 Conjecture variant, in which edges must be assigned labels from dedicated lists of three labels, and the Multiplicative 1-2-3 Conjecture variant, in which labels 1, 2, 3 must be assigned to the edges so that adjacent vertices are incident to different products of labels. Several results obtained towards these two variants led to observe some behaviours that are distant from those of the original conjecture. In this work, we consider the list version of the Multiplicative 1-2-3 Conjecture, proposing the first study dedicated to this very problem. In particular, given any graph G, we wonder about the minimum k such that G can be labelled as desired when its edges must be assigned labels from dedicated lists of size k. Exploiting a relationship between our problem and the List 1-2-3 Conjecture, we provide upper bounds on k when G belongs to particular classes of graphs. We further improve some of these bounds through dedicated arguments. PubDate: 2022-05-04

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Abstract: Abstract In this paper, we analyze embeddings of grid graphs on orientable surfaces. We determine the genus of two infinite classes of 3-dimensional grid graphs that do not admit quadrilateral embeddings and effective upper bounds for the genus of any 3-dimensional grid graph, both in terms of a grid graph’s combinatorics. As an application, we provide a complete classification of planar and toroidal grid graphs. Our work requires a variety of combinatorial and graph theoretic arguments to determine effective lower bounds on the genus of a grid graph, along with explicitly constructing embeddings of grid graphs on surfaces to determine effective upper bounds on their genera. PubDate: 2022-04-16

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Abstract: Abstract Let \(\lambda \) be a positive integer. An acyclic \(\lambda \) -coloring of a digraph D is a partition of the vertices of D into \(\lambda \) color clases such that the color classes induce acyclic subdigraphs in D. The minimum integer \(\lambda \) for which there exists an acyclic \(\lambda \) -coloring of D is the dichromatic number dc(D) of D. Let \(P(D;\lambda )\) be the dichromatic polynomial of D, which is the number of acyclic \(\lambda \) -colorings of D. In this paper, a recursive formula for \(P(D;\lambda )\) is given. The coefficients of the polynomial \(P(D;\lambda )\) are studied. The dichromatic polynomial of a digraph D is related to the structure of its underlying graph UG(D). Also, we study dichromatic equivalently and dichromatically unique digraphs. PubDate: 2022-04-16

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Abstract: Abstract We address the problem of constructing large circulant networks with given degree and diameter, and efficient routing schemes. First we discuss the theoretical upper bounds and their asymptotics. Then we apply concepts and tools from the change-making problem to efficient routing in circulant graphs. With these tools we investigate some of the families of circulant graphs that have been proposed in the literature, and we construct tables of large circulant graphs and digraphs with efficient routing properties. PubDate: 2022-04-16

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Abstract: Abstract A graph is called H-free if it does not contain H as a subgraph. The Turán number of H, denoted by ex(n, H), is the maximum number of edges in any H-free graph on n vertices. For sufficiently large n, Lidický et al. (Electron J Combin 20:62, 2013) determined ex(n, F), where F denotes a class of forest with components each of order 4, and they characterized all the extremal graphs. Moreover, Lan et al. (Appl Math Comput 348: 270–274, 2019) proved that the extremal graph is unique when n is large enough. Motivated by their results, we consider a class of forests F with each component a path or star of order 6, we determine ex(n, F) for sufficiently large n and we also characterize all the extremal graphs. Furthermore, we prove that the extremal graph is unique when n is large enough. Let \(kP_n\) denote the disjoint union of k copies of the path \(P_n\) on n vertices. Recently, Lan et al. (Discuss Math Graph Theory 39: 805–814, 2019) determined the exact value of \(ex(n,2P_7)\) . Motivated by their result, we show that \(ex(n,2P_9) = max\{{(7n+153+r(r-8))/2},7n-27\}\) for all \(n\ge 18\) , where r is the remainder of \(n-17\) when divided by 8. PubDate: 2022-04-15

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Abstract: Abstract Inductive constructions are established for countably infinite simple graphs which have minimally rigid locally generic placements in \({{\mathbb{R}}}^2\) . This generalises a well-known result of Henneberg for generically rigid finite graphs. Inductive methods are also employed in the determination of the infinitesimal flexibility dimension of countably infinite graphs associated with infinitely faceted convex polytopes in \({{\mathbb{R}}}^3\) . In particular, a generalisation of Cauchy’s rigidity theorem is obtained. PubDate: 2022-04-13

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Abstract: Abstract In this paper, using matrix techniques, we compute the Ihara-zeta function and the number of spanning trees of the join of two semi-regular bipartite graphs. Furthermore, we show that the spectrum and the Ihara-zeta function of the join of two semi-regular bipartite graphs can determine each other. PubDate: 2022-04-07

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Abstract: Abstract Rödl, Ruciński, and Szemerédi determined the minimum \((k-1)\) -degree threshold for the existence of fractional perfect matchings in k-uniform hypergrahs, and Kühn, Osthus, and Townsend extended this result by asymptotically determining the d-degree threshold for the range \(k-1>d\ge k/2\) . In this note, we prove the following exact degree threshold: let k, d be positive integers with \(k\ge 4\) and \(k-1>d\ge k/2\) , and let n be any integer with \(n\ge 2k(k-1)+1\) . Then any n-vertex k-uniform hypergraph with minimum d-degree \(\delta _d(H)>{n-d\atopwithdelims ()k-d} -{n-d-(\lceil n/k\rceil -1)\atopwithdelims ()k-d}\) contains a fractional perfect matching. This lower bound on the minimum d-degree is best possible. We also determine the minimum d-degree threshold for the existence of fractional matchings of size s, where \(0<s\le n/k\) (when \(k/2\le d\le k-1\) ), or with s large enough and \(s\le n/k\) (when \(2k/5<d<k/2\) ). PubDate: 2022-04-06

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Abstract: Abstract Given a positive integer r, let \([r]=\{1,\ldots ,r\}\) . Let n, m be positive integers such that n is sufficiently large and \(1\le m\le \lfloor n/3\rfloor -1\) . Let H be a 3-graph with vertex set [n], and let \(\delta _1(H)\) denote the minimum vertex degree of H. The size of a maximum matching of H is denoted by \(\nu (H)\) . Kühn, Osthus and Treglown (2013) proved that there exists an integer \(n_0\in \mathbb {N}\) such that if H is a 3-graph with \(n\ge n_0\) vertices and \(\delta _1(H)>{n-1\atopwithdelims ()2}-{n-m\atopwithdelims ()2}\) , then \(\nu (H)\ge m\) . In this paper, we show that there exists an integer \(n_1\in \mathbb {N}\) such that if \( V(H) \ge n_1\) , \(\delta _1(H)>{n-1\atopwithdelims ()2}-{n-m\atopwithdelims ()2}+3\) and \(\nu (H)\le m\) , then H is a subgraph of \(H^*(n,m)\) , where \(H^*(n,m)\) is a 3-graph with vertex set [n] and edge set \(E(H^*(n,m))=\{e\subseteq [n]: e =3 \text{ and } e\cap [m] \ne \emptyset \}\) . The minimum degree condition is best possible. PubDate: 2022-04-06