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《Discrete Mathematics》2021,344(12):112622
A Deza graph G with parameters is a k-regular graph with n vertices such that any two distinct vertices have b or a common neighbours. The children and of a Deza graph G are defined on the vertex set of G such that every two distinct vertices are adjacent in or if and only if they have a or b common neighbours, respectively. A strongly Deza graph is a Deza graph with strongly regular children. In this paper we give a spectral characterisation of strongly Deza graphs, show relationships between eigenvalues, and study strongly Deza graphs which are distance-regular. 相似文献
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John Asplund Kossi Edoh Ruth Haas Yulia Hristova Beth Novick Brett Werner 《Discrete Mathematics》2018,341(10):2938-2948
For a graph and, the shortest path reconfiguration graph of with respect to and is denoted by . The vertex set of is the set of all shortest paths between and in . Two vertices in are adjacent, if their corresponding paths in differ by exactly one vertex. This paper examines the properties of shortest path graphs. Results include establishing classes of graphs that appear as shortest path graphs, decompositions and sums involving shortest path graphs, and the complete classification of shortest path graphs with girth 5 or greater. We include an infinite family of well structured examples, showing that the shortest path graph of a grid graph is an induced subgraph of a lattice. 相似文献
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For an integer , a graph is -hamiltonian if for any vertex subset with , is hamiltonian, and is -hamiltonian connected if for any vertex subset with , is hamiltonian connected. Thomassen in 1984 conjectured that every 4-connected line graph is hamiltonian (see Thomassen, 1986), and Ku?zel and Xiong in 2004 conjectured that every 4-connected line graph is hamiltonian connected (see Ryjá?ek and Vrána, 2011). In Broersma and Veldman (1987), Broersma and Veldman raised the characterization problem of -hamiltonian line graphs. In Lai and Shao (2013), it is conjectured that for , a line graph is -hamiltonian if and only if is -connected. In this paper we prove the following.(i) For an integer , the line graph of a claw-free graph is -hamiltonian if and only if is -connected.(ii) The line graph of a claw-free graph is 1-hamiltonian connected if and only if is 4-connected. 相似文献
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In this article, we obtain a sufficient condition related to toughness for a graph to be all fractional -critical. We prove that if for some nonnegative integers , then is all fractional -critical. Our result improves the known results in Liu and Zhang (2008) and Liu and Cai (2009). 相似文献
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《Discrete Mathematics》2020,343(12):112117
Let be an edge-colored graph of order . The minimum color degree of , denoted by , is the largest integer such that for every vertex , there are at least distinct colors on edges incident to . We say that an edge-colored graph is rainbow if all its edges have different colors. In this paper, we consider vertex-disjoint rainbow triangles in edge-colored graphs. Li (2013) showed that if , then contains a rainbow triangle and the lower bound is tight. Motivated by this result, we prove that if and , then contains two vertex-disjoint rainbow triangles. In particular, we conjecture that if , then contains vertex-disjoint rainbow triangles. For any integer , we show that if and , then contains vertex-disjoint rainbow triangles. Moreover, we provide sufficient conditions for the existence of edge-disjoint rainbow triangles. 相似文献
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A -list assignment of a graph is a mapping that assigns to each vertex a list of at least colors satisfying for each edge . A graph is -choosable if there exists an -coloring of for every -list assignment . This concept is also known as choosability with separation. In this paper, we prove that any planar graph is -choosable if any -cycle is not adjacent to a -cycle, where and . 相似文献
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《Discrete Mathematics》2022,345(8):112917
Let and denote the flow number and the circular flow number of a flow-admissible signed graph , respectively. It is known that for every unsigned graph G. Based on this fact, in 2011 Raspaud and Zhu conjectured that holds also for every flow-admissible signed graph . This conjecture was disproved by Schubert and Steffen using graphs with bridges and vertices of large degree. In this paper we focus on cubic graphs, since they play a crucial role in many open problems in graph theory. For cubic graphs we show that if and only if and if , then . We also prove that all pairs of flow number and circular flow number that fulfil these conditions can be achieved in the family of bridgeless cubic graphs and thereby disprove the conjecture of Raspaud and Zhu even for bridgeless signed cubic graphs. Finally, we prove that all currently known flow-admissible graphs without nowhere-zero 5-flow have flow number and circular flow number 6 and propose several conjectures in this area. 相似文献
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For a graph , the -dominating graph of has vertices corresponding to the dominating sets of having cardinality at most , where two vertices of are adjacent if and only if the dominating set corresponding to one of the vertices can be obtained from the dominating set corresponding to the second vertex by the addition or deletion of a single vertex. We denote the domination and upper domination numbers of by and , respectively, and the smallest integer for which is connected for all by . It is known that , but constructing a graph such that appears to be difficult.We present two related constructions. The first construction shows that for each integer and each integer such that , there exists a graph such that , and . The second construction shows that for each integer and each integer such that , there exists a graph such that , and . 相似文献
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An incidence of a graph is a pair where is a vertex of and is an edge of incident to . Two incidences and of are adjacent whenever (i) , or (ii) , or (iii) or . An incidence-coloring of is a mapping from the set of incidences of to a set of colors such that every two adjacent incidences receive distinct colors. The notion of incidence coloring has been introduced by Brualdi and Quinn Massey (1993) from a relation to strong edge coloring, and since then, has attracted a lot of attention by many authors.On a list version of incidence coloring, it was shown by Benmedjdoub et al. (2017) that every Hamiltonian cubic graph is incidence 6-choosable. In this paper, we show that every cubic (loopless) multigraph is incidence 6-choosable. As a direct consequence, it implies that the list strong chromatic index of a -bipartite graph is at most 6, where a (2,3)-bipartite graph is a bipartite graph such that one partite set has maximum degree at most 2 and the other partite set has maximum degree at most 3. 相似文献
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A graph G is called a pseudo-core if every endomorphism of G is either an automorphism or a colouring. A graph G is a core if every endomorphism of G is an automorphism. Let be the finite field with q elements where q is a power of an odd prime number. The quadratic forms graph, denoted by where , has all quadratic forms on as vertices and two vertices f and g are adjacent whenever or 2. We prove that every is a pseudo-core. Further, when n is even, is a core. When n is odd, is not a core. On the other hand, we completely determine the independence number of . 相似文献
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Christian Bosse 《Discrete Mathematics》2019,342(12):111595
The Hadwiger number of a graph , denoted , is the largest integer such that contains as a minor. A famous conjecture due to Hadwiger in 1943 states that for every graph , , where denotes the chromatic number of . Let denote the independence number of . A graph is -free if it does not contain the graph as an induced subgraph. In 2003, Plummer, Stiebitz and Toft proved that for all -free graphs with , where is any graph on four vertices with , , or is a particular graph on seven vertices. In 2010, Kriesell subsequently generalized the statement to include all forbidden subgraphs on five vertices with . In this note, we prove that for all -free graphs with , where denotes the wheel on six vertices. 相似文献
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《Discrete Mathematics》2019,342(2):339-343
A strong edge-coloring of a graph is a partition of its edge set into induced matchings. Let be a connected planar graph with girth and maximum degree . We show that either is isomorphic to a subgraph of a very special -regular graph with girth , or has a strong edge-coloring using at most colors. 相似文献
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