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1.
《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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《Discrete Mathematics》2022,345(1):112668
The following optimal stopping problem is considered. The vertices of a graph G are revealed one by one, in a random order, to a selector. He aims to stop this process at a time t that maximizes the expected number of connected components in the graph , induced by the currently revealed vertices. The selector knows G in advance, but different versions of the game are considered depending on the information that he gets about . We show that when G has N vertices and maximum degree of order , then the number of components of is concentrated around its mean, which implies that playing the optimal strategy the selector does not benefit much by receiving more information about . Results of similar nature were previously obtained by M. Lasoń for the case where G is a k-tree (for constant k). We also consider the particular cases where G is a square, triangular or hexagonal lattice, showing that an optimal selector gains cN components and we compute c with an error less than 0.005 in each case. 相似文献
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《Discrete Mathematics》2022,345(11):113036
Let G be a cyclically 5-connected cubic graph with a 5-edge-cut separating G into two cyclic components and . We prove that each component can be completed to a cyclically 5-connected cubic graph by adding three vertices, unless is a cycle of length five. Our work extends similar results by Andersen et al. for cyclic connectivity 4 from 1988. 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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Let G be a finite, connected graph. The average distance of a vertex v of G is the arithmetic mean of the distances from v to all other vertices of G. The remoteness and the proximity of G are the maximum and the minimum of the average distances of the vertices of G. In this paper, we present a sharp upper bound on the remoteness of a triangle-free graph of given order and minimum degree, and a corresponding bound on the proximity, which is sharp apart from an additive constant. We also present upper bounds on the remoteness and proximity of -free graphs of given order and minimum degree, and we demonstrate that these are close to being best possible. 相似文献
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《Discrete Mathematics》2019,342(5):1471-1480
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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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Aleksander Vesel 《Discrete Mathematics》2019,342(4):1139-1146
The generalized Fibonacci cube is the graph obtained from the -cube by removing all vertices that contain a given binary string as a substring. If is an induced subgraph of , then the cube-complement of is the graph induced by the vertices of which are not in . In particular, the cube-complement of a generalized Fibonacci cube is the subgraph of induced by the set of all vertices that contain as a substring. The questions whether a cube-complement of a generalized Fibonacci cube is (i) connected, (ii) an isometric subgraph of a hypercube or (iii) a median graph are studied. Questions (ii) and (iii) are completely solved, i.e. the sets of binary strings that allow a graph of this class to be an isometric subgraph of a hypercube or a median graph are given. The cube-complement of a daisy cube is also studied. 相似文献
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《Discrete Mathematics》2022,345(4):112784
A set S of vertices in a graph G is a dominating set if every vertex not in S is adjacent to a vertex in S. If, in addition, every vertex in S is adjacent to some other vertex in S, then S is a total dominating set. The domination number of G is the minimum cardinality of a dominating set in G, while the total domination number of G is the minimum cardinality of total dominating set in G. A claw-free graph is a graph that does not contain as an induced subgraph. Let G be a connected, claw-free, cubic graph of order n. We show that if we exclude two graphs, then , and this bound is best possible. In order to prove this result, we prove that if we exclude four graphs, then , and this bound is best possible. These bounds improve previously best known results due to Favaron and Henning (2008) [7], Southey and Henning (2010) [19]. 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
13.
Ruwen Hollenbach 《Journal of Pure and Applied Algebra》2022,226(1):106781
Let G be a simple, simply connected algebraic group of exceptional type defined over with Frobenius endomorphism . Let be a good prime for G. We determine the number of irreducible Brauer characters in the quasi-isolated ?-blocks of . This is done by proving that generalized e-Harish-Chandra theory holds for the Lusztig series associated to quasi-isolated elements of . 相似文献
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《Discrete Mathematics》2022,345(3):112706
The power of a graph , , is the graph whose vertex set is V and in which two distinct vertices are adjacent if and only if their distance in G is at most k. This article proves various eigenvalue bounds for the independence number and chromatic number of which purely depend on the spectrum of G, together with a method to optimize them. Our bounds for the k-independence number also work for its quantum counterpart, which is not known to be a computable parameter in general, thus justifying the use of integer programming to optimize them. Some of the bounds previously known in the literature follow as a corollary of our main results. Infinite families of graphs where the bounds are sharp are presented as well. 相似文献
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Given a subgroup of the symmetric group , the cycle index polynomial is the average of the power-sum symmetric polynomials indexed by the cycle types of permutations in . By Pólya’s Theorem, the monomial expansion of is the generating function for weighted colorings of objects, where we identify colorings related by one of the symmetries in . This paper develops combinatorial formulas for the fundamental quasisymmetric expansions and Schur expansions of certain cycle index polynomials. We give explicit bijective proofs based on standardization algorithms applied to equivalence classes of colorings. Subgroups studied here include Young subgroups of , the alternating groups , direct products, conjugate subgroups, and certain cyclic subgroups of generated by . The analysis of these cyclic subgroups when is prime reveals an unexpected connection to perfect matchings on a hypercube with certain vertices identified. 相似文献
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《Discrete Mathematics》2019,342(4):927-933
Given a fixed positive integer , the -planar local crossing number of a graph , denoted by , is the minimum positive integer such that can be decomposed into subgraphs, each of which can be drawn in a plane such that no edge is crossed more than times. In this note, we show that under certain natural restrictions, the ratio is of order , which is analogous to the result of Pach et al. (2018) for the -planar crossing number (defined as the minimum positive integer for which there is a -planar drawing of with total edge crossings). As a corollary of our proof we show that, under similar restrictions, one may obtain a -planar drawing of with both the total number of edge crossings as well as the maximum number of times any edge is crossed essentially matching the best known bounds. Our proof relies on the crossing number inequality and several probabilistic tools such as concentration of measure and the Lovász local lemma. 相似文献
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《Discrete Mathematics》2022,345(8):112902
For a simple graph G, denote by n, , and its order, maximum degree, and chromatic index, respectively. A graph G is edge-chromatic critical if and for every proper subgraph H of G. Let G be an n-vertex connected regular class 1 graph, and let be obtained from G by splitting one vertex of G into two vertices. Hilton and Zhao in 1997 conjectured that must be edge-chromatic critical if , and they verified this when . In this paper, we prove it for . 相似文献