共查询到20条相似文献,搜索用时 15 毫秒
1.
In 1995, Voigt constructed a planar triangle-free graph that is not 3-list-colorable. It has 166 vertices. Gutner then constructed such a graph with 164 vertices. We present two more graphs with these properties. The first graph has 97 vertices and a failing list assignment using triples from a set of six colors, while the second has 109 vertices and a failing list assignment using triples from a set of five colors. 相似文献
2.
3.
This paper proves the following result. Assume is a triangle-free planar graph, is an independent set of . If is a list assignment of such that for each vertex and for each vertex , then is -colorable. 相似文献
4.
A weighted graph (G,w) is a graph G together with a positive weight-function on its vertex set w : V(G)→R>0. The weighted domination number γw(G) of (G,w) is the minimum weight w(D)=∑vDw(v) of a set DV(G) such that every vertex xV(D)−D has a neighbor in D. If ∑vV(G)w(v)=|V(G)|, then we speak of a normed weighted graph. Recently, we proved thatfor normed weighted bipartite graphs (G,w) of order n such that neither G nor the complement
has isolated vertices. In this paper we will extend these Nordhaus–Gaddum-type results to triangle-free graphs. 相似文献
5.
Independent domination in triangle-free graphs 总被引:1,自引:0,他引:1
Julie Haviland 《Discrete Mathematics》2008,308(16):3545-3550
Let G be a simple graph of order n and minimum degree δ. The independent domination numberi(G) is defined to be the minimum cardinality among all maximal independent sets of vertices of G. We establish upper bounds, as functions of n and δ?n/2, for the independent domination number of triangle-free graphs, and over part of the range achieve best possible results. 相似文献
6.
7.
Thomassen recently proved, using the Tutte cycle technique, that if G is a 3-connected cubic triangle-free planar graph then G contains a bipartite subgraph with at least edges, improving the previously known lower bound . We extend Thomassen’s technique and further improve this lower bound to . 相似文献
8.
Albert Guan 《Discrete Mathematics》2009,309(20):6044-6047
Given a (possibly improper) edge colouring F of a graph G, a vertex colouring of G is adapted toF if no colour appears at the same time on an edge and on its two endpoints. A graph G is called (for some positive integer k) if for any list assignment L to the vertices of G, with |L(v)|≥k for all v, and any edge colouring F of G, G admits a colouring c adapted to F where c(v)∈L(v) for all v. This paper proves that a planar graph G is adaptably 3-choosable if any two triangles in G have distance at least 2 and no triangle is adjacent to a 4-cycle. 相似文献
9.
We show that any orientation of a graph with maximum degree three has an oriented 9-colouring, and that any orientation of a graph with maximum degree four has an oriented 69-colouring. These results improve the best known upper bounds of 11 and 80, respectively. 相似文献
10.
Xuding Zhu 《Discrete Mathematics》2009,309(18):5562-5568
Given a graph G and a positive integer p, χp(G) is the minimum number of colours needed to colour the vertices of G so that for any i≤p, any subgraph H of G of tree-depth i gets at least i colours. This paper proves an upper bound for χp(G) in terms of the k-colouring number of G for k=2p−2. Conversely, for each integer k, we also prove an upper bound for in terms of χk+2(G). As a consequence, for a class K of graphs, the following two statements are equivalent:
- (a)
- For every positive integer p, χp(G) is bounded by a constant for all G∈K.
- (b)
- For every positive integer k, is bounded by a constant for all G∈K.
- (c)
- For every positive integer q, ∇q(G) (the greatest reduced average density of G with rank q) is bounded by a constant for all G∈K.
11.
In order to avoid interference in cellular telephone networks, sets of radio frequencies are to be assigned to transmitters such that adjacent transmitters are allotted disjoint sets of frequencies. Often these transmitters are laid out like vertices of a triangular lattice in a plane. This problem corresponds to the problem of multicoloring an induced subgraph of a triangular lattice with integer demands associated with each vertex. We deal with the simpler case of triangle-free subgraphs of the lattice. [Frédéric Havet, Discrete Math. 233 (2001) 1–3] uses inductive arguments to prove that triangle-free hexagonal graphs can be colored with colors where ωd is the maximum demand on a clique in the graph. We give a simpler proof and hope that our techniques can be used to prove the conjecture by [McDiarmid and Reed, Networks Suppl. 36 (2000) 114–117] that these graphs are -multicolorable. 相似文献
12.
Colin McDiarmid 《Annals of Operations Research》1984,1(3):183-200
We discuss some results concerned with the behaviour of colouring algorithms on large random graphs. 相似文献
13.
A signed graph has a plus or minus sign on each edge. A simple cycle is positive or negative depending on whether it contains an even or odd number of negative edges, respectively. We consider embeddings of a signed graph in the projective plane for which a simple cycle is essential if and only if it is negative. We characterize those signed graphs that have such a projective-planar embedding. Our characterization is in terms of a related signed graph formed by considering the theta subgraphs in the given graph. 相似文献
14.
15.
The Catalan numbers occur in various counting problems in combinatorics. This paper reveals a connection between the Catalan numbers and list colouring of graphs. Assume is a graph and is a mapping. For a nonnegative integer , let be the extension of to the graph for which for each vertex of . Let be the minimum such that is not -choosable and be the minimum such that is not -paintable. We study the parameter and for arbitrary mappings . For , an -dominated path ending at is a monotonic path of the grid from to such that each vertex on satisfies . Let be the number of -dominated paths ending at . By this definition, the Catalan number equals . This paper proves that if has vertices and , then , where and for . Therefore, if , then equals the Catalan number . We also show that if is the disjoint union of graphs and , then and . This generalizes a result in Carraher et al. (2014), where the case each is a copy of is considered. 相似文献
16.
We prove that intersection graphs of boxes on the plane with girth 6 and 8 are 3- and 2-degenerate, respectively. This implies that these graphs are 4- and 3-list-colourable, respectively. 相似文献
17.
18.
Konrad K. Dabrowski François Dross Matthew Johnson Daniël Paulusma 《Journal of Graph Theory》2019,92(4):377-393
A colouring of a graph is a function such that for every . A -regular list assignment of is a function with domain such that for every , is a subset of of size . A colouring of respects a -regular list assignment of if for every . A graph is -choosable if for every -regular list assignment of , there exists a colouring of that respects . We may also ask if for a given -regular list assignment of a given graph , there exists a colouring of that respects . This yields the -Regular List Colouring problem. For , we determine a family of classes of planar graphs, such that either -Regular List Colouring is -complete for instances with , or every is -choosable. By using known examples of non--choosable and non--choosable graphs, this enables us to classify the complexity of -Regular List Colouring restricted to planar graphs, planar bipartite graphs, planar triangle-free graphs, and planar graphs with no -cycles and no -cycles. We also classify the complexity of -Regular List Colouring and a number of related colouring problems for graphs with bounded maximum degree. 相似文献
19.
20.
A graph is -colorable if it admits a vertex partition into a graph with maximum degree at most and a graph with maximum degree at most . We show that every -free planar graph is -colorable. We also show that deciding whether a -free planar graph is -colorable is NP-complete. 相似文献