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A digraph T is strong if for every pair of vertices u and v there exists a directed path from u to v and a directed path from v to u. Denote the in-degree and out-degree of a vertex v of T by d-(v) and d+(v), respectively. We define δ-(T)=minvV(T){d-(v)} and δ+(T)=minvV(T){d+(v)}. Let T0 be a 7-tournament which contains no transitive 4-subtournament. In this paper, we obtain some conditions on a strong tournament which cannot be partitioned into two cycles. We show that a strong tournament T with n6 vertices such that TT0 and max{δ+(T),δ-(T)}3 can be partitioned into two cycles. Finally, we give a sufficient condition for a tournament to be partitioned into k cycles. 相似文献
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《Discrete Mathematics》2022,345(1):112672
Recently, Andrews proved two conjectures of Beck related to ranks of partitions. Very recently, Chern established some results on weighted rank and crank moments and proved many Andrews-Beck type congruences. Motivated by Andrews and Chern's work, Lin, Peng and Toh proved a number of Andrews-Beck type congruences for k-colored partitions. At the end of their paper, Lin, Peng and Toh posed several conjectures on Andrews-Beck type congruences. In this paper, we confirm one of those conjectures based on some q-series identities. 相似文献
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David C. Fisher 《Journal of Graph Theory》1996,23(1):43-48
Let the square of a tournament be the digraph on the same nodes with arcs where the directed distance in the tournament is at most two. This paper verifies Dean's conjecture: any tournament has a node whose outdegree is at least doubled in its square. © 1996 John Wiley & Sons, Inc. 相似文献
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In this paper we prove a conjecture of Metsch about the maximum number of lines intersecting a pointset in PG(2,q), presented at the conference “Combinatorics 2002”. As a consequence, we give a short proof of the famous Jamison, Brouwer and Schrijver bound on the size of the smallest affine blocking set in AG(2,q). 相似文献
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For a graph G, let σ2(G) denote the minimum degree sum of a pair of nonadjacent vertices. We conjecture that if |V(G)| = n = Σki = 1 ai and σ2(G) ≥ n + k − 1, then for any k vertices v1, v2,…, vk in G, there exist vertex‐disjoint paths P1, P2,…, Pk such that |V(Pi)| = ai and vi is an endvertex of Pi for 1 ≤ i ≤ k. In this paper, we verify the conjecture for the cases where almost all ai ≤ 5, and the cases where k ≤ 3. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 163–169, 2000 相似文献
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Shuya Chiba 《Discrete Mathematics》2018,341(10):2912-2918
For a vertex subset of a graph , let be the maximum value of the degree sums of the subsets of of size . In this paper, we prove the following result: Let be positive integers, and let be an -connected graph of order . If for every independent set of size in , then has a 2-factor with exactly cycles. This is a common extension of the results obtained by Brandt et al. (1997) and Yamashita (2008), respectively. 相似文献
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Proof of a conjecture of Fiedler and Markham 总被引:4,自引:0,他引:4
Xuerong Yong 《Linear algebra and its applications》2000,320(1-3):167-171
Let A be an n×n nonsingular M-matrix. For the Hadamard product AA−1, M. Fiedler and T.L. Markham conjectured in [Linear Algebra Appl. 10l (1988) 1] that q(AA−1)2/n, where q(AA−1) is the smallest eigenvalue (in modulus) of AA−1. We considered this conjecture in [Linear Algebra Appl. 288 (1999) 259] having observed an incorrect proof in [Linear Algebra Appl. 144 (1991) 171] and obtained that q(AA−1)(2/n)(n−1)/n. The present paper gives a proof for this conjecture. 相似文献
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A tree T is said to be bad, if it is the vertex‐disjoint union of two stars plus an edge joining the center of the first star to an end‐vertex of the second star. A tree T is good, if it is not bad. In this article, we prove a conjecture of Alan Hartman that, for any spanning tree T of K2m, where m ≥ 4, there exists a (2m − 1)‐edge‐coloring of K2m such that all the edges of T receive distinct colors if and only if T is good. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 7–17, 1999 相似文献
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O. Favaron H. Karami R. Khoeilar S. M. Sheikholeslami L. Volkmann 《Journal of Graph Theory》2010,64(4):323-329
Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A b‐dimensional cube is a Cartesian product I1×I2×···×Ib, where each Ii is a closed interval of unit length on the real line. The cubicity of G, denoted by cub(G), is the minimum positive integer b such that the vertices in G can be mapped to axis parallel b‐dimensional cubes in such a way that two vertices are adjacent in G if and only if their assigned cubes intersect. An interval graph is a graph that can be represented as the intersection of intervals on the real line—i.e. the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. Suppose S(m) denotes a star graph on m+1 nodes. We define claw number ψ(G) of the graph to be the largest positive integer m such that S(m) is an induced subgraph of G. It can be easily shown that the cubicity of any graph is at least ?log2ψ(G)?. In this article, we show that for an interval graph G ?log2ψ(G)??cub(G)??log2ψ(G)?+2. It is not clear whether the upper bound of ?log2ψ(G)?+2 is tight: till now we are unable to find any interval graph with cub(G)>?log2ψ(G)?. We also show that for an interval graph G, cub(G)??log2α?, where α is the independence number of G. Therefore, in the special case of ψ(G)=α, cub(G) is exactly ?log2α2?. The concept of cubicity can be generalized by considering boxes instead of cubes. A b‐dimensional box is a Cartesian product I1×I2×···×Ib, where each Ii is a closed interval on the real line. The boxicity of a graph, denoted box(G), is the minimum k such that G is the intersection graph of k‐dimensional boxes. It is clear that box(G)?cub(G). From the above result, it follows that for any graph G, cub(G)?box(G)?log2α?. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 323–333, 2010 相似文献
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Let e be a positive integer, p be an odd prime, , and be the finite field of q elements. Let . The graph is a bipartite graph with vertex partitions and , and edges defined as follows: a vertex is adjacent to a vertex if and only if and . If and , the graph contains no cycles of length less than eight and is edge-transitive. Motivated by certain questions in extremal graph theory and finite geometry, people search for examples of graphs containing no cycles of length less than eight and not isomorphic to the graph , even without requiring them to be edge-transitive. So far, no such graphs have been found. It was conjectured that if both f and g are monomials, then no such graphs exist. In this paper we prove the conjecture. 相似文献
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Tam��s Terpai 《Combinatorica》2011,31(6):739-754
Using analytical tools, we prove that for any simple graph G on n vertices and its complement [`(G)]\bar G the inequality $\mu \left( G \right) + \mu \left( {\bar G} \right) \leqslant \tfrac{4}
{3}n - 1$\mu \left( G \right) + \mu \left( {\bar G} \right) \leqslant \tfrac{4}
{3}n - 1 holds, where μ(G) and m( [`(G)] )\mu \left( {\bar G} \right) denote the greatest eigenvalue of adjacency matrix of the graphs G and [`(G)]\bar G respectively. 相似文献
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Walter Bergweiler 《Mathematische Zeitschrift》1990,204(1):381-390
Research performed as a Feodor Lynen Research Fellow of the Alexander von Humboldt Foundation at Cornell University 相似文献
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Aimo Tietäväinen 《Journal of Number Theory》1975,7(3):353-356
Let θ(k, p) be the least s such that the congruence has a nontrivial solution. Let θ(k) = {max θ(k, p)| p > 1 + 2k}. The purpose of this note is to prove the following conjecture of S. Chowla: . 相似文献
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Rotation symmetric Boolean functions have important applications in the design of cryptographic algorithms. We prove the conjecture about rotation symmetric Boolean functions (RSBFs) of degree 3 proposed in Cusick and St?nic? (2002) [2] and its generalization, thus the nonlinearity of such functions is determined. 相似文献