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The tensor product of graphs , and is defined by and Let be the fractional chromatic number of a graph . In this paper, we prove that if one of the three graphs , and is a circular clique, 相似文献
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In the two disjoint shortest paths problem ( 2-DSPP), the input is a graph (or a digraph) and its vertex pairs and , and the objective is to find two vertex-disjoint paths and such that is a shortest path from to for , if they exist. In this paper, we give a first polynomial-time algorithm for the undirected version of the 2-DSPP with an arbitrary non-negative edge length function. 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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《Discrete Mathematics》2019,342(5):1275-1292
A discrete function of variables is a mapping , where , and are arbitrary finite sets. Function is called separable if there exist functions for , such that for every input the function takes one of the values . Given a discrete function , it is an interesting problem to ask whether is separable or not. Although this seems to be a very basic problem concerning discrete functions, the complexity of recognition of separable discrete functions of variables is known only for . In this paper we will show that a slightly more general recognition problem, when is not fully but only partially defined, is NP-complete for . We will then use this result to show that the recognition of fully defined separable discrete functions is NP-complete for .The general recognition problem contains the above mentioned special case for . This case is well-studied in the context of game theory, where (separable) discrete functions of variables are referred to as (assignable) -person game forms. There is a known sufficient condition for assignability (separability) of two-person game forms (discrete functions of two variables) called (weak) total tightness of a game form. This property can be tested in polynomial time, and can be easily generalized both to higher dimension and to partially defined functions. We will prove in this paper that weak total tightness implies separability for (partially defined) discrete functions of variables for any , thus generalizing the above result known for . Our proof is constructive. Using a graph-based discrete algorithm we show how for a given weakly totally tight (partially defined) discrete function of variables one can construct separating functions in polynomial time with respect to the size of the input function. 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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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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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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《Discrete Mathematics》2022,345(9):112945
The coinvariant algebra is a quotient of the polynomial ring whose algebraic properties are governed by the combinatorics of permutations of length n. A word over the positive integers is packed if whenever appears as a letter of w, so does . We introduce a quotient of which is governed by the combinatorics of packed words. We relate our quotient to the generalized coinvariant rings of Haglund, Rhoades, and Shimozono as well as the superspace coinvariant ring. 相似文献
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Jakub Przybyło 《Discrete Mathematics》2019,342(2):498-504
Let be any graph without isolated edges. The well known 1–2–3 Conjecture asserts that the edges of can be weighted with so that adjacent vertices have distinct weighted degrees, i.e. the sums of their incident weights. It was independently conjectured that if additionally has no isolated triangles, then it can be edge decomposed into two subgraphs which fulfil the 1–2–3 Conjecture with just weights 1,2, i.e. such that there exist weightings so that for every , if then , where denotes the sum of weights incident with in for . We apply the probabilistic method to prove that the known weakening of this so-called Standard (2,2)-Conjecture holds for graphs with minimum degree large enough. Namely, we prove that if , then can be decomposed into graphs for which weightings exist so that for every , or . In fact we prove a stronger result, as one of the weightings is redundant, i.e. uses just weight 1. 相似文献
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