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 Discussiones Mathematicae Graph TheoryJournal Prestige (SJR): 0.633 Citation Impact (citeScore): 1Number of Followers: 1     Open Access journal ISSN (Print) 1234-3099 - ISSN (Online) 2083-5892 Published by Sciendo  [370 journals]
• On the Equality of Domination Number and 2-Domination Number

• Abstract: The 2-domination number γ2(G) of a graph G is the minimum cardinality of a set D ⊆ V (G) for which every vertex outside D is adjacent to at least two vertices in D. Clearly, γ2(G) cannot be smaller than the domination number γ(G). We consider a large class of graphs and characterize those members which satisfy γ2 = γ. For the general case, we prove that it is NP-hard to decide whether γ2 = γ holds. We also give a necessary and sufficient condition for a graph to satisfy the equality hereditarily.
PubDate: Tue, 22 Mar 2022 00:00:00 GMT

• Edge Degree Conditions for Dominating and Spanning Closed Trails

• Abstract: Edge degree conditions have been studied since the 1980s, mostly with regard to hamiltonicity of line graphs and the equivalent existence of dominating closed trails in their root graphs, as well as the stronger property of being supereulerian, i.e., admitting a spanning closed trail. For a graph G, let σ¯2(G)=min{ d(u)+d(v) uv∈E(G) }{\bar \sigma _2}\left( G \right) = \min \left\{ {d\left( u \right) + d\left( v \right) uv \in E\left( G \right)} \right\}. Chen et al. conjectured that a 3-edge-connected graph G with sufficientl large order n and σ¯2(G)>n9-2{\bar \sigma _2}\left( G \right) > {n \over 9} - 2 is either supereulerian or contractible to the Petersen graph. We show that the conjecture is true when σ¯2(G)≥2(⌊ n/15 ⌋−1){\bar \sigma _2}\left( G \right) \ge 2\left( {\left\lfloor {n/15} \right\rfloor - 1} \right). Furthermore, we show that for an essentially k-edge-connected graph G with sufficiently large order n, the following statements hold.(i) If k = 2 and σ¯2(G)≥2(⌊ n/8 ⌋−1){\bar \sigma _2}\left( G \right) \ge 2\left( {\left\lfloor {n/8} \right\rfloor - 1} \right), then either L(G) is hamiltonian or G can be contracted to one of a set of six graphs which are not supereulerian;(ii) If k = 3 and σ¯2(G)≥2(⌊ n/15 ⌋−1){\bar \sigma _2}\left( G \right) \ge 2\left( {\left\lfloor {n/15} \right\rfloor - 1} \right), then either L(G) is hamiltonian or G can be contracted to the Petersen graph.
PubDate: Tue, 22 Mar 2022 00:00:00 GMT

• The Achromatic Number of the Cartesian Product of and

• Abstract: Let G be a graph and C a finite set of colours. A vertex colouring f : V (G) → C is complete if for any pair of distinct colours c1, c2 ∈ C one can find an edge {v1, v2} ∈ E(G) such that f(vi) = ci, i = 1, 2. The achromatic number of G is defined to be the maximum number achr(G) of colours in a proper complete vertex colouring of G. In the paper achr(K6 □ Kq) is determined for any integer q such that either 8 ≤ q ≤ 40 or q ≥ 42 is even.
PubDate: Tue, 22 Mar 2022 00:00:00 GMT

• Chorded -Pancyclic and Weakly -Pancyclic Graphs

• Abstract: As natural relaxations of pancyclic graphs, we say a graph G is kpancyclic if G contains cycles of each length from k to V (G) and G is weakly pancyclic if it contains cycles of all lengths from the girth to the circumference of G, while G is weakly k-pancyclic if it contains cycles of all lengths from k to the circumference of G. A cycle C is chorded if there is an edge between two vertices of the cycle that is not an edge of the cycle. Combining these ideas, a graph is chorded pancyclic if it contains chorded cycles of each length from 4 to the circumference of the graph, while G is chorded k-pancyclic if there is a chorded cycle of each length from k to V (G) . Further, G is chorded weakly k-pancyclic if there is a chorded cycle of each length from k to the circumference of the graph. We consider conditions for graphs to be chorded weakly k-pancyclic and chorded k-pancyclic.
PubDate: Fri, 18 Feb 2022 00:00:00 GMT

• On -Hamiltonian-Connected Line Graphs

• Abstract: For an integer s ≥ 0, G is s-hamiltonian-connected if for any vertex subset S ⊆ V (G) with S ≤ s, G − S is hamiltonian-connected. Thomassen in 1984 conjectured that every 4-connected line graph is hamiltonian (see [Reflections on graph theory, J. Graph Theory 10 (1986) 309–324]), and Kužel and Xiong in 2004 conjectured that every 4-connected line graph is hamiltonian-connected (see [Z. Ryjáček and P. Vrána, Line graphs of multigraphs and Hamilton-connectedness of claw-free graphs, J. Graph Theory 66 (2011) 152–173]). In this paper we prove the following. (i) For s ≥ 3, every (s + 4)-connected line graph is s-hamiltonian-connected. (ii) For s ≥ 0, every (s + 4)-connected line graph of a claw-free graph is s-hamiltonian-connected.
PubDate: Fri, 18 Feb 2022 00:00:00 GMT

• Cubic Graphs Having Only -Cycles in Each 2-Factor

• Abstract: We consider the class of 2-connected cubic graphs having only k-cycles in each 2-factor, and obtain the following two results: (i) every 2-connected cubic graph having only 8-cycles in each 2-factor is isomorphic to a unique Hamiltonian graph of order 8; and (ii) a 2-connected cubic planar graph G has only k-cycles in each 2-factor if and only if k = 4 and G is the complete graph of order 4.
PubDate: Fri, 18 Feb 2022 00:00:00 GMT

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