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《Discrete Mathematics》2022,345(9):112942
A graph G is k-degenerate if every subgraph of G has a vertex with degree at most k. Using the Euler's formula, one can obtain that planar graphs without 3-cycles are 3-degenerate. Wang and Lih, and Fijav? et al. proved the analogue results for planar graphs without 5-cycles and planar graphs without 6-cycles, respectively. Recently, Liu et al. showed that planar graphs without 3-cycles adjacent to 5-cycles are 3-degenerate. In this work, we generalized all aforementioned results by showing that planar graphs without mutually adjacent 3-,5-, and 6-cycles are 3-degenerate. A graph G without mutually adjacent 3-,5-, and 6-cycles means that G cannot contain three graphs, say , and , where is a 3-cycle, is a 5-cycle, and is a 6-cycle such that each pair of , and are adjacent. As an immediate consequence, we have that every planar graph without mutually adjacent 3-,5-, and 6-cycles is DP-4-colorable. This consequence also generalizes the result by Chen et al that planar graphs without 5-cycles adjacent to 6-cycles are DP-4-colorable. 相似文献
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《Discrete Mathematics》2022,345(12):113069
The toughness of a noncomplete graph G is the maximum real number t such that the ratio of to the number of components of is at least t for every cutset S of G. Determining the toughness for a given graph is NP-hard. Chvátal's toughness conjecture, stating that there exists a constant such that every graph with toughness at least is hamiltonian, is still open for general graphs. A graph is called -free if it does not contain any induced subgraph isomorphic to , the disjoint union of and two isolated vertices. In this paper, we confirm Chvátal's toughness conjecture for -free graphs by showing that every 7-tough -free graph on at least three vertices is hamiltonian. 相似文献
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Let be the -color Ramsey number of an odd cycle of length . It is shown that for each fixed , for all sufficiently large , where is a constant. This improves an old result by Bondy and Erd?s (1973). 相似文献
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《Discrete Mathematics》2019,342(4):1017-1027
We study the independence number of a product of Kneser graph with itself, where we consider all four standard graph products. The cases of the direct, the lexicographic and the strong product of Kneser graphs are not difficult (the formula for is presented in this paper), while the case of the Cartesian product of Kneser graphs is much more involved. We establish a lower bound and an upper bound for the independence number of , which are asymptotically tending to and , respectively. The former is obtained by a construction, which differs from the standard diagonalization procedure, while for the upper bound the -independence number of Kneser graphs can be applied. We also establish some constructions in odd graphs , which give a lower bound for the 2-independence number of these graphs, and prove that two such constructions give the same lower bound as a previously known one. Finally, we consider the -stable Kneser graphs , derive a formula for their -independence number, and give the exact value of the independence number of the Cartesian square of . 相似文献
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A graph is -colorable if it admits a vertex partition into a graph with maximum degree at most and a graph with maximum degree at most . We show that every -free planar graph is -colorable. We also show that deciding whether a -free planar graph is -colorable is NP-complete. 相似文献
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Define a -star to be the complete bipartite graph . In a 2014 article, Hoffman and Roberts prove that a partial -star decomposition of can be embedded in a -star decomposition of where is at most if is odd and if is even. In our work, we offer a straightforward construction for embedding partial -star designs and lower these bounds to and , respectively. 相似文献
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《Discrete Mathematics》2022,345(1):112659
In a recent paper, Gerbner, Patkós, Tuza and Vizer studied regular F-saturated graphs. One of the essential questions is given F, for which n does a regular n-vertex F-saturated graph exist. They proved that for all sufficiently large n, there is a regular -saturated graph with n vertices. We extend this result to both and and prove some partial results for larger complete graphs. Using a variation of sum-free sets from additive combinatorics, we prove that for all , there is a regular -saturated with n vertices for infinitely many n. Studying the sum-free sets that give rise to -saturated graphs is an interesting problem on its own and we state an open problem in this direction. 相似文献
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For a positive integer , a graph is -knitted if for each subset of vertices, and every partition of into (disjoint) parts for some , one can find disjoint connected subgraphs such that contains for each . In this article, we show that if the minimum degree of an -vertex graph is at least when , then is -knitted. The minimum degree is sharp. As a corollary, we obtain that -contraction-critical graphs are -connected. 相似文献
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The Erd?s–Gallai Theorem states that every graph of average degree more than contains a path of order for . In this paper, we obtain a stability version of the Erd?s–Gallai Theorem in terms of minimum degree. Let be a connected graph of order and be disjoint paths of order respectively, where , , and . If the minimum degree , then except several classes of graphs for sufficiently large , which extends and strengths the results of Ali and Staton for an even path and Yuan and Nikiforov for an odd path. 相似文献
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For a connected amply regular graph with parameters satisfying , it is known that its diameter is bounded by . This was generalized by Terwilliger to -graphs satisfying . It follows from Terwilliger that a connected amply regular graph with parameters satisfying and has diameter at most 7.In this paper we will classify the 2-walk-regular graphs with valency and diameter at least 4 such that its intersection number satisfies . This result generalizes a result of Koolen and Park for distance-regular graphs. And we show that if such a 2-walk-regular graph is not distance-regular, then it is the incidence graph of a group divisible design with the dual property with parameters . 相似文献