List Coloring Of Graphs Research Articles

A note on improper DP-colouring of planar graphs

ABSTRACT DP-colouring (also known as correspondence colouring), introduced by Dvořák and Postle, is a generalization of list colouring. Many results on list-colouring of graphs, especially of planar graphs, have been extended to the setting of DP-colouring. Recently, Pongpat and Kittikorn [P. Sittitrai and K. Nakprasit, Suffficient conditions on planar graphs to have a relaxed DP-3-colourability, Graphs and Combinatorics 35 (2019), pp. 837–845.] introduced DP--colouring to generalize -colouring and -choosability. They proved that every planar graph G without -cycles is DP--colourable. In this note, we show the following results:(1) Every planar graph G without -cycles is DP--colourable; (2) Every planar graph G without -cycles is DP--colourable; (3) Every planar graph G without -cycles is DP--colourable.

Minimum choosability of planar graphs

The problem of list coloring of graphs appears in practical problems concerning channel or frequency assignment. In this paper, we study the minimum number of choosability of planar graphs. A graph G is edge-k-choosable if whenever every edge x is assigned with a list of at least k colors, L(x)), there exists an edge coloring $$\phi $$ such that $$\phi (x) \in L(x)$$ . Similarly, A graph G is toal-k-choosable if when every element (edge or vertex) x is assigned with a list of at least k colors, L(x), there exists an total coloring $$\phi $$ such that $$\phi (x) \in L(x)$$ . We proved $$\chi '_{l}(G)=\Delta $$ and $$\chi ''_{l}(G)=\Delta +1$$ for a planar graph G with maximum degree $$\Delta \ge 8$$ and without chordal 6-cycles, where the list edge chromatic number $$\chi '_{l}(G)$$ of G is the smallest integer k such that G is edge-k-choosable and the list total chromatic number $$\chi ''_{l}(G)$$ of G is the smallest integer k such that G is total-k-choosable.

含有交叉数的图的在线列表染色

含有交叉数的图的在线列表染色

List colouring of graphs and generalized Dyck paths

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 G is a graph and f:V(G)→N is a mapping. For a nonnegative integer m, let f(m) be the extension of f to the graph G▪ Km¯ for which f(m)(v)=|V(G)| for each vertex v of Km¯. Let mc(G,f) be the minimum m such that G▪ Km¯ is not f(m)-choosable and mp(G,f) be the minimum m such that G▪ Km¯ is not f(m)-paintable. We study the parameter mc(Kn,f) and mp(Kn,f) for arbitrary mappings f. For x→=(x1,x2,…,xn), an x→-dominated path ending at (a,b) is a monotonic path P of the a×b grid from (0,0) to (a,b) such that each vertex (i,j) on P satisfies i≤xj+1. Let ψ(x→) be the number of x→-dominated paths ending at (xn,n). By this definition, the Catalan number Cn equals ψ((0,1,…,n−1)). This paper proves that if G=Kn has vertices v1,v2,…,vn and f(v1)≤f(v2)≤…≤f(vn), then mc(G,f)=mp(G,f)=ψ(x→(f)), where x→(f)=(x1,x2,…,xn) and xi=f(vi)−i for i=1,2,…,n. Therefore, if f(vi)=n, then mc(Kn,f)=mp(Kn,f) equals the Catalan number Cn. We also show that if G=G1∪G2∪⋯∪Gp is the disjoint union of graphs G1,G2,…,Gp and f=f1∪f2∪⋯∪fp, then mc(G,f)=∏i=1pmc(Gi,fi) and mp(G,f)=∏i=1pmp(Gi,fi). This generalizes a result in Carraher et al. (2014), where the case each Gi is a copy of K1 is considered.

On (4, 2)‐Choosable Graphs

AbstractA graph G is called ‐choosable if for any list assignment L that assigns to each vertex v a set of a permissible colors, there is a b‐tuple L‐coloring of G. An (a, 1)‐choosable graph is also called a‐choosable. In the pioneering article on list coloring of graphs by Erdős et al. , 2‐choosable graphs are characterized. Confirming a special case of a conjecture in , Tuza and Voigt proved that 2‐choosable graphs are ‐choosable for any positive integer m. On the other hand, Voigt proved that if m is an odd integer, then these are the only ‐choosable graphs; however, when m is even, there are ‐choosable graphs that are not 2‐choosable. A graph is called 3‐choosable‐critical if it is not 2‐choosable, but all its proper subgraphs are 2‐choosable. Voigt conjectured that for every positive integer m, all bipartite 3‐choosable‐critical graphs are ‐choosable. In this article, we determine which 3‐choosable‐critical graphs are (4, 2)‐choosable, refuting Voigt's conjecture in the process. Nevertheless, a weaker version of the conjecture is true: we prove that there is an even integer k such that for any positive integer m, every bipartite 3‐choosable‐critical graph is ‐choosable. Moving beyond 3‐choosable‐critical graphs, we present an infinite family of non‐3‐choosable‐critical graphs that have been shown by computer analysis to be (4, 2)‐choosable. This shows that the family of all (4, 2)‐choosable graphs has rich structure.

List coloring of graphs having cycles of length divisible by a given number

Let G be a graph and χ l ( G ) denote the list chromatic number of G . In this paper we prove that for every graph G for which the length of each cycle is divisible by l ( l ≥ 3 ), χ l ( G ) ≤ 3 .

Extending Hall's Theorem into List Colorings: A Partial History

In 1988, A. J. W. Hilton and P. D. Johnson Jr. found a natural generalization of the condition in Philip Hall's celebrated theorem on systems of distinct representatives. This generalization was formed in the relatively new theory of list colorings of graphs. Here we give an account of a strand of development arising from this generalization, concentrating on extensions of Hall's theorem. New results are presented concerning list colorings of independence systems and colorings of graphs with nonnegative measurable functions on positive measure spaces.

Circular list colorings of some graphs

The circular list coloring is a circular version of list colorings of graphs. Let χinc,l denote the circular choosability(or the circular list chromatic number). In this paper, the circular choosability of outer planar graphs and odd wheel is discussed.

On connected list colorings of graphs

Vertex colorings of graphs and edge colorings of multigraphs are studied such that elements of each color induce a connected subgraph. The relationship is examined between arising quantitative parameters and other known parameters.

Oriented list colorings of graphs

Oriented list colorings of graphs

List-colourings of graphs

In this paper, we go some way towards proving a conjecture of Albertson and Tucker. Among other results, we show that ifc>11/6 and Δ is sufficiently large, then a graph of maximum degree Δ with a list of cardinality [cΔ] assigned to each edge may be edge-coloured so that each edge is coloured with an element of its list.