共查询到20条相似文献，搜索用时 15 毫秒
1.
Jan van den Heuvel Daniel Král' Martin Kupec Jean‐Sébastien Sereni Jan Volec 《Journal of Graph Theory》2014,77(4):299-329
We study the following problem: given a real number k and an integer d, what is the smallest ε such that any fractional ‐precoloring of vertices at pairwise distances at least d of a fractionally k‐colorable graph can be extended to a fractional ‐coloring of the whole graph? The exact values of ε were known for and any d. We determine the exact values of ε for if , and if , and give upper bounds for if , and if . Surprisingly, ε viewed as a function of k is discontinuous for all those values of d. 相似文献
2.
Petr A. Golovach Matthew Johnson Daniël Paulusma Jian Song 《Journal of Graph Theory》2017,84(4):331-363
For a positive integer k, a k‐coloring of a graph is a mapping such that whenever . The Coloring problem is to decide, for a given G and k, whether a k‐coloring of G exists. If k is fixed (i.e., it is not part of the input), we have the decision problem k‐Coloring instead. We survey known results on the computational complexity of Coloring and k‐Coloring for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial coloring, or where lists of permissible colors are given for each vertex. 相似文献
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Dniel Marx 《Journal of Graph Theory》2005,49(4):313-324
In the edge precoloring extension problem, we are given a graph with some of the edges having preassigned colors and it has to be decided whether this coloring can be extended to a proper k‐edge‐coloring of the graph. In list edge coloring every edge has a list of admissible colors, and the question is whether there is a proper edge coloring where every edge receives a color from its list. We show that both problems are NP‐complete on (a) planar 3‐regular bipartite graphs, (b) bipartite outerplanar graphs, and (c) bipartite series‐parallel graphs. This improves previous results of Easton and Parker 6 , and Fiala 8 . © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 313–324, 2005 相似文献
5.
Let G=(V, E) be a graph where every vertex v∈V is assigned a list of available colors L(v). We say that G is list colorable for a given list assignment if we can color every vertex using its list such that adjacent vertices get different colors. If L(v)={1, …, k} for all v∈V then a corresponding list coloring is nothing other than an ordinary k‐coloring of G. Assume that W?V is a subset of V such that G[W] is bipartite and each component of G[W] is precolored with two colors taken from a set of four. The minimum distance between the components of G[W] is denoted by d(W). We will show that if G is K4‐minor‐free and d(W)≥7, then such a precoloring of W can be extended to a 4‐coloring of all of V. This result clarifies a question posed in 10. Moreover, we will show that such a precoloring is extendable to a list coloring of G for outerplanar graphs, provided that |L(v)|=4 for all v∈V\W and d(W)≥7. In both cases the bound for d(W) is best possible. © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 284‐294, 2009 相似文献
6.
Hajo Broersma Fedor V. Fomin Petr A. Golovach Gerhard J. Woeginger 《Journal of Graph Theory》2007,55(2):137-152
We introduce and study backbone colorings, a variation on classical vertex colorings: Given a graph G = (V,E) and a spanning subgraph H of G (the backbone of G), a backbone coloring for G and H is a proper vertex coloring V → {1,2,…} of G in which the colors assigned to adjacent vertices in H differ by at least two. We study the cases where the backbone is either a spanning tree or a spanning path. We show that for tree backbones of G the number of colors needed for a backbone coloring of G can roughly differ by a multiplicative factor of at most 2 from the chromatic number χ(G); for path backbones this factor is roughly . We show that the computational complexity of the problem “Given a graph G, a spanning tree T of G, and an integer ?, is there a backbone coloring for G and T with at most ? colors?” jumps from polynomial to NP‐complete between ? = 4 (easy for all spanning trees) and ? = 5 (difficult even for spanning paths). We finish the paper by discussing some open problems. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 137–152, 2007 相似文献
7.
B. Ries 《Discrete Applied Mathematics》2010,158(5):592-596
In this note we consider two coloring problems in mixed graphs, i.e., graphs containing edges and arcs, which arise from scheduling problems where disjunctive and precedence constraints have to be taken into account. We show that they are both NP-complete in cubic planar bipartite mixed graphs, which strengthens some results of Ries and de Werra (2008) [9]. 相似文献
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On 3-colorability of planar graphs without adjacent short cycles 总被引：1，自引：0，他引：1
WANG YingQian MAO XiangHua LU HuaJing & WANG WeiFan College of Mathematics Physics Information Engineering Zhejiang Normal University Jinhua China College of Basic Science Ningbo Dahongying University Ningbo 《中国科学 数学(英文版)》2010,(4)
A short cycle means a cycle of length at most 7.In this paper,we prove that planar graphs without adjacent short cycles are 3-colorable.This improves a result of Borodin et al.(2005). 相似文献
10.
Dániel Marx 《Operations Research Letters》2005,33(4):382-384
In the minimum sum coloring problem we have to assign positive integers to the vertices of a graph in such a way that neighbors receive different numbers and the sum of the numbers is minimized. Szkalicki has shown that minimum sum coloring is NP-hard for interval graphs. Here we present a simpler proof of this result. 相似文献
11.
Ji?í Fiala 《Discrete Applied Mathematics》2010,158(7):771-368
Packing coloring is a partitioning of the vertex set of a graph with the property that vertices in the i-th class have pairwise distance greater than i. The main result of this paper is a solution of an open problem of Goddard et al. showing that the decision whether a tree allows a packing coloring with at most k classes is NP-complete.We further discuss specific cases when this problem allows an efficient algorithm. Namely, we show that it is decideable in polynomial time for graphs of bounded treewidth and diameter, and fixed parameter tractable for chordal graphs.We accompany these results by several observations on a closely related variant of the packing coloring problem, where the lower bounds on the distances between vertices inside color classes are determined by an infinite nondecreasing sequence of bounded integers. 相似文献
12.
Adrian Bondy Guillermo Durán Min Chih Lin Jayme L. Szwarcfiter 《Journal of Graph Theory》2003,44(3):178-192
The clique graph of a graph is the intersection graph of its (maximal) cliques. A graph is self-clique when it is isomorphic with its clique graph, and is clique-Helly when its cliques satisfy the Helly property. We prove that a graph is clique-Helly and self-clique if and only if it admits a quasi-symmetric clique matrix, that is, a clique matrix whose families of row and column vectors are identical. We also give a characterization of such graphs in terms of vertex-clique duality. We describe new classes of self-clique and 2-self-clique graphs. Further, we consider some problems on permuted matrices (matrices obtained by permuting the rows and/or columns of a given matrix). We prove that deciding whether a (0,1)-matrix admits a symmetric (quasi-symmetric) permuted matrix is graph (hypergraph) isomorphism complete. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 178–192, 2003 相似文献
13.
Paul Bonsma 《Journal of Graph Theory》2009,62(2):109-126
The Matching‐Cut problem is the problem to decide whether a graph has an edge cut that is also a matching. Previously this problem was studied under the name of the Decomposable Graph Recognition problem, and proved to be ‐complete when restricted to graphs with maximum degree four. In this paper it is shown that the problem remains ‐complete for planar graphs with maximum degree four, answering a question by Patrignani and Pizzonia. It is also shown that the problem is ‐complete for planar graphs with girth five. The reduction is from planar graph 3‐colorability and differs from earlier reductions. In addition, for certain graph classes polynomial time algorithms to find matching‐cuts are described. These classes include claw‐free graphs, co‐graphs, and graphs with fixed bounded tree‐width or clique‐width. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 109–126, 2009 相似文献
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The odd‐girth of a graph is the length of a shortest odd circuit. A conjecture by Pavol Hell about circular coloring is solved in this article by showing that there is a function ƒ(ϵ) for each ϵ : 0 < ϵ < 1 such that, if the odd‐girth of a planar graph G is at least ƒ(ϵ), then G is (2 + ϵ)‐colorable. Note that the function ƒ(ϵ) is independent of the graph G and ϵ → 0 if and only if ƒ(ϵ) → ∞. A key lemma, called the folding lemma, is proved that provides a reduction method, which maintains the odd‐girth of planar graphs. This lemma is expected to have applications in related problems. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 109–119, 2000 相似文献
15.
A graph G is k‐choosable if its vertices can be colored from any lists L(ν) of colors with |L(ν)| ≥ k for all ν ∈ V(G). A graph G is said to be (k,?)‐choosable if its vertices can be colored from any lists L(ν) with |L(ν)| ≥k, for all ν∈ V(G), and with . For each 3 ≤ k ≤ ?, we construct a graph G that is (k,?)‐choosable but not (k,? + 1)‐choosable. On the other hand, it is proven that each (k,2k ? 1)‐choosable graph G is O(k · ln k · 24k)‐choosable. © 2005 Wiley Periodicals, Inc. J Graph Theory 相似文献
16.
The star chromatic index of a graph G is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi‐colored. We obtain a near‐linear upper bound in terms of the maximum degree . Our best lower bound on in terms of Δ is valid for complete graphs. We also consider the special case of cubic graphs, for which we show that the star chromatic index lies between 4 and 7 and characterize the graphs attaining the lower bound. The proofs involve a variety of notions from other branches of mathematics and may therefore be of certain independent interest. 相似文献
17.
In this article, we introduce the new notion of acyclic improper colorings of graphs. An improper coloring of a graph is a vertex-coloring in which adjacent vertices are allowed to have the same color, but each color class Vi satisfies some condition depending on i. Such a coloring is acyclic if there are no alternating 2-colored cycles. We prove that every outerplanar graph can be acyclically 2-colored in such a way that each monochromatic subgraph has degree at most five and that this result is best possible. For planar graphs, we prove some negative results and state some open problems. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 97–107, 1999 相似文献
18.
Given lists of available colors assigned to the vertices of a graph G, a list coloring is a proper coloring of G such that the color on each vertex is chosen from its list. If the lists all have size k, then a list coloring is equitable if each color appears on at most vertices. A graph is equitably k-choosable if such a coloring exists whenever the lists all have size k. We prove that G is equitably k-choosable when unless G contains or k is odd and . For forests, the threshold improves to . If G is a 2-degenerate graph (given k ≥ 5) or a connected interval graph (other than ), then G is equitably k-choosable when . © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 166–177, 2003 相似文献
19.
Michael J. Pelsmajer 《Journal of Graph Theory》2009,61(2):127-139
Given lists of available colors assigned to the vertices of a graph G, a list coloring is a proper coloring of G such that the color on each vertex is chosen from its list. If the lists all have size k, then a list coloring is equitable if each color appears on at most ?|V(G)|/k? vertices. A graph is equitably k ‐choosable if such a coloring exists whenever the lists all have size k. Kostochka, Pelsmajer, and West introduced this notion and conjectured that G is equitably k‐choosable for k>Δ(G). We prove this for graphs of treewidth w≤5 if also k≥3w?1. We also show that if G has treewidth w≥5, then G is equitably k‐choosable for k≥max{Δ(G)+w?4, 3w?1}. As a corollary, if G is chordal, then G is equitably k‐choosable for k≥3Δ(G)?4 when Δ(G)>2. © 2009 Wiley Periodicals, Inc. J Graph Theory 相似文献
20.
David Galvin 《Journal of Graph Theory》2013,73(1):66-84
For graphs G and H, a homomorphism from G to H, or H‐coloring of G, is an adjacency preserving map from the vertex set of G to the vertex set of H. Our concern in this article is the maximum number of H‐colorings admitted by an n‐vertex, d‐regular graph, for each H. Specifically, writing for the number of H‐colorings admitted by G, we conjecture that for any simple finite graph H (perhaps with loops) and any simple finite n‐vertex, d‐regular, loopless graph G, we have where is the complete bipartite graph with d vertices in each partition class, and is the complete graph on vertices.Results of Zhao confirm this conjecture for some choices of H for which the maximum is achieved by . Here, we exhibit for the first time infinitely many nontrivial triples for which the conjecture is true and for which the maximum is achieved by .We also give sharp estimates for and in terms of some structural parameters of H. This allows us to characterize those H for which is eventually (for all sufficiently large d) larger than and those for which it is eventually smaller, and to show that this dichotomy covers all nontrivial H. Our estimates also allow us to obtain asymptotic evidence for the conjecture in the following form. For fixed H, for all d‐regular G, we have where as . More precise results are obtained in some special cases. 相似文献