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《Discrete Mathematics》2019,342(9):2632-2635
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We view an undirected graph as a symmetric digraph, where each edge is replaced by two opposite arcs and . Assume is an inverse closed subset of permutations of positive integers. We say is --colourable if for any mapping with , there is a mapping such that for each arc . The concept of --colourable is a common generalization of several other colouring concepts. This paper is focused on finding the sets such that every triangle-free planar graph is -3-colourable. Such a set is called TFP-good. Grötzsch’s theorem is equivalent to say that is TFP-good. We prove that for any inverse closed subset of which is not isomorphic to , is TFP-good if and only if either or there exists such that for each , . It remains an open question to determine whether or not is TFP-good. 相似文献
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I. Allie 《Discrete Mathematics》2019,342(2):387-392
Let be a bridgeless cubic graph. Oddness (weak oddness) is defined as the minimum number of odd components in a 2-factor (an even factor) of , denoted as (Steffen, 2004) ( Lukot’ka and Mazák (2016)). Oddness and weak oddness have been referred to as measurements of uncolourability (Fiol et al., 2017, Lukot’ka and Mazák, 2016, Lukot’ka et al., 2015 and, Steffen, 2004), due to the fact that and if and only if is 3-edge-colourable. Another so-called measurement of uncolourability is resistance, defined as the minimum number of edges that can be removed from such that the resulting graph is 3-edge-colourable, denoted as (Steffen, 2004). It is easily shown that . While it has been shown that the difference between any two of these measures can be arbitrarily large, it has been conjectured that , and that if is a snark then (Fiol et al., 2017). In this paper, we disprove the latter by showing that the ratio of oddness to weak oddness can be arbitrarily large. We also offer some insights into the former conjecture by defining what we call resistance reducibility, and hypothesizing that almost all cubic graphs are such resistance reducible. 相似文献
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《Discrete Mathematics》2021,344(12):112604
A well-known theorem of Vizing states that if G is a simple graph with maximum degree Δ, then the chromatic index of G is Δ or . A graph G is class 1 if , and class 2 if ; G is Δ-critical if it is connected, class 2 and for every . A long-standing conjecture of Vizing from 1968 states that every Δ-critical graph on n vertices has at least edges. We initiate the study of determining the minimum number of edges of class 1 graphs G, in addition, for every . Such graphs have intimate relation to -co-critical graphs, where a non-complete graph G is -co-critical if there exists a k-coloring of such that G does not contain a monochromatic copy of but every k-coloring of contains a monochromatic copy of for every . We use the bound on the size of the aforementioned class 1 graphs to study the minimum number of edges over all -co-critical graphs. We prove that if G is a -co-critical graph on vertices, then where ε is the remainder of when divided by 2. This bound is best possible for all and . 相似文献
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《Discrete Mathematics》2020,343(1):111640
For any graph with , a shortest path reconfiguration graph can be formed with respect to and ; we denote such a graph as . The vertex set of is the set of all shortest paths from to in while two vertices in are adjacent if and only if the vertex sets of the paths that represent and differ in exactly one vertex. In a recent paper (Asplund et al., 2018), it was shown that shortest path graphs with girth five or greater are exactly disjoint unions of even cycles and paths. In this paper, we extend this result by classifying all shortest path graphs with no induced 4-cycles. 相似文献
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《Discrete Mathematics》2021,344(12):112600
An -colored-mixed graph is a graph having m colors of arcs and n colors of edges. We do not allow two arcs or edges to have the same endpoints. A homomorphism from an -colored-mixed graph G to another -colored-mixed graph H is a morphism such that each edge (resp. arc) of G is mapped to an edge (resp. arc) of H of the same color (and orientation). An -colored-mixed graph T is said to be -universal if every graph in (the planar -colored-mixed graphs with girth at least g) admits a homomorphism to T.We show that planar -universal graphs do not exist for (and any value of g) and find a minimal (in the number vertices) planar -universal graphs in the other cases. 相似文献
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《Discrete Mathematics》2019,342(10):2770-2782
“Which graphs are determined by their spectrum (DS for short)?” is a fundamental question in spectral graph theory. It is generally very hard to show a given graph to be DS and few results about DS graphs are known in literature. In this paper, we consider the above problem in the context of the generalized -spectrum. A graph is said to be determined by the generalized -spectrum (DGQS for short) if, for any graph , and have the same -spectrum and so do their complements imply that is isomorphic to . We give a simple arithmetic condition for a graph being DGQS. More precisely, let be a graph with adjacency matrix and degree diagonal matrix . Let be the signless Laplacian matrix of , and ( is the all-ones vector) be the -walk matrix. We show that if (which is always an integer) is odd and square-free, then is DGQS. 相似文献
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Let be a simple graph, and let , and denote the maximum degree, the average degree and the chromatic index of , respectively. We called edge--critical if and for every proper subgraph of . Vizing in 1968 conjectured that if is an edge--critical graph of order , then . We prove that for any edge--critical graph , that is, This result improves the best known bound obtained by Woodall in 2007 for . 相似文献
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A biased graph is a graph with a class of selected circles (“cycles”, “circuits”), called balanced, such that no theta subgraph contains exactly two balanced circles. A biased graph has two natural matroids, the frame matroid and the lift matroid , and their extensions the full frame matroid and the extended (or complete) lift matroid . In Part IV we used algebra to study the representations of these matroids by vectors over a skew field and the corresponding embeddings in Desarguesian projective spaces. Here we redevelop those representations, independently of Part IV and in greater generality, by using synthetic geometry. 相似文献
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Let be a non-negative real number, and let be the largest eigenvalue of . Specially, and are called the spectral radius and signless Laplacian spectral radius of , respectively. A graph is said to be Hamiltonian (traceable) if it contains a Hamiltonian cycle (path), and a graph is called Hamilton-connected if any two vertices are connected by a Hamiltonian path in . The number of edges of is denoted by . Recently, the (signless Laplacian) spectral property of Hamiltonian (traceable, Hamilton-connected) graphs received much attention. In this paper, we shall give a general result for all these existed results. To do this, we first generalize the concept of Hamiltonian, traceable, and Hamilton-connected to -suitable, and we secondly present a lower bound for to confirm the existence of -suitable graphs. Thirdly, when , we obtain a lower bound for to confirm the existence of -suitable graphs. Consequently, our results generalize and improve all these existed results in this field, including the main results of Chen et al. (2018), Feng et al. (2017), Füredi et al. (2017), Ge et al. (2016), Li et al. (2016), Nikiforov et al. (2016), Wei et al. (2019) and Yu et al. (2013, 2014). 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