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《Discrete Mathematics》2022,345(11):113065
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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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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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《Discrete Mathematics》2022,345(8):112903
Graphs considered in this paper are finite, undirected and loopless, but we allow multiple edges. The point partition number is the least integer k for which G admits a coloring with k colors such that each color class induces a -degenerate subgraph of G. So is the chromatic number and is the point arboricity. The point partition number with was introduced by Lick and White. A graph G is called -critical if every proper subgraph H of G satisfies . In this paper we prove that if G is a -critical graph whose order satisfies , then G can be obtained from two non-empty disjoint subgraphs and by adding t edges between any pair of vertices with and . Based on this result we establish the minimum number of edges possible in a -critical graph G of order n and with , provided that and t is even. For the corresponding two results were obtained in 1963 by Tibor Gallai. 相似文献
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Let be an array of nonnegative numbers satisfying the recurrence relation with and unless . In this paper, we first prove that the array can be generated by some context-free Grammars, which gives a unified proof of many known results. Furthermore, we present criteria for real rootedness of row-generating functions and asymptotical normality of rows of . Applying the criteria to some arrays related to tree-like tableaux, interior and left peaks, alternating runs, flag descent numbers of group of type , and so on, we get many results in a unified manner. Additionally, we also obtain the continued fraction expansions for generating functions related to above examples. As results, we prove the strong -log-convexity of some generating functions. 相似文献
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For given graphs , , the -color Ramsey number, denoted by , is the smallest integer such that if we arbitrarily color the edges of a complete graph of order with colors, then it always contains a monochromatic copy of colored with , for some . Let be a cycle of length and a star of order . In this paper, firstly we give a general upper bound of . In particular, for the 3-color case, we have and this bound is tight in some sense. Furthermore, we prove that for all and , and if is a prime power, then the equality holds. 相似文献
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Michael Skotnica 《Discrete Mathematics》2019,342(12):111611
Let denote the maximal number of points on the discrete torus (discrete toric grid) of sizes with no three collinear points. The value is known for the case where is prime. It is also known that . In this paper we generalize some of the known tools for determining and also show some new. Using these tools we prove that the sequence is periodic for all fixed . In general, we do not know the period; however, if for prime, then we can bound it. We prove that which implies that the period for the sequence is , where is at most . 相似文献
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In 1996, Cox and Rodger [Cycle systems of the line graph of the complete graph, J. Graph Theory 21 (1996) 173–182] raised the following question: For what values of and does there exist an -cycle decomposition of In this paper, the above question is answered for In fact, it is shown that the -fold line graph of the complete graph has a -decomposition if and only if and 相似文献
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Let be a simple connected graph with vertices and edges. The spectral radius of is the largest eigenvalue of its adjacency matrix. In this paper, we firstly consider the effect on the spectral radius of a graph by removing a vertex, and then as an application of the result, we obtain a new sharp upper bound of which improves some known bounds: If , where is an integer, then The equality holds if and only if is a complete graph or , where is the graph obtained from by deleting some edge . 相似文献
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Let and be positive integers with . Given a permutation of integers , we consider -consecutive sums of , i.e., for , where we let . What we want to do in this paper is to know the exact value of where denotes the set of all permutations of . In this paper, we determine the exact values of for some particular cases of and . As a corollary of the results, we obtain , and for any . 相似文献
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《Stochastic Processes and their Applications》2020,130(4):2127-2158
Let be an ergodic diffusion with invariant distribution . Consider the empirical measure where is an Euler scheme with decreasing steps which approximates . Given a test function , we obtain sharp concentration inequalities for which improve the results in Honoré et al. (2019). Our hypotheses on the test function cover many real applications: either is supposed to be a coboundary of the infinitesimal generator of the diffusion, or is supposed to be Lipschitz. 相似文献