We study online algorithms with advice for the problem of coloring graphs which come as input vertex by vertex. We consider the class of all 3-colorable graphs and its sub-classes of chordal and maximal outerplanar graphs, respectively. We show that, in the case of the first two classes, for coloring optimally, essentially log2 3 advice bits per vertex (bpv) are necessary and sufficient. In the...

A proper vertex coloring of a non oriented graph G = (V, E) is linear if the graph induced by the vertices of two color classes is a forest of paths. A graph G is L-list colorable if for a given list assignment L = {L(v) : v ∈ V }, there exists a proper coloring c of G such that c(v) ∈ L(v) for all v ∈ V . If G is L-list colorable for every list assignment with |L(v)| ≥ k for all v ∈ V , then G...

Given a non-decreasing sequence S = (s1, s2, . . . , sk) of positive integers, an Spacking edge-coloring of a graph G is a partition of the edge set of G into k subsets {X1,X2, . . . ,Xk} such that for each 1 ≤ i ≤ k, the distance between two distinct edges e, e′ ∈ Xi is at least si + 1. This paper studies S-packing edgecolorings of cubic graphs. Among other results, we prove that cubic graphs ...

A graph G is called interval colorable if it has a proper edge coloring with colors 1,2,3,… such that the of edges incident to every vertex form an integers. Not all graphs are colorable; in fact, quite few families have been proved admit colorings. In this paper we introduce and investigate new notion, thickness G, denoted θint(G), which minimum number edge-disjoint subgraphs whose union G. Ou...

In the previous paper, it was shown that the graph U. ÷ 1 obtained from the wheel W n ÷ 1 by deleting a spoke is uniquely determined by its chromatic polynomial if n >i 3 is odd. In this paper, we show that the result is also true for even n >~ 4 except when n = 6 in which case, the graph W given in the paper is the only graph having the same chromatic polynomial as that of U 7. The relevant to...

Dvo?ák and Postle introduced the concept of DP-coloring to overcome some difficulties in list coloring. Sittitrai Nakprasit combined defective coloring define a new coloring—relaxed DP-coloring. For relaxed DP-coloring, Sribunhung et al. proved that planar graphs without 4- 7-cycles are DP-(0, 2, 2)-colorable. Li 4, 8-cycles or 9-cycles DP-(1, 1, 1)-colorable. Lu Zhu 5-cycles, 6-cycles, In this...

A Fary grid drawing of a graph is a drawing on a three-dimensional grid such that vertices are placed at integer coordinates and edges are straight-lines such that no edge crossings are allowed. In this paper it is proved that each k-colorable graph (k 2) needs at least (n 3=2) volume to be drawn. Furthermore, it is shown how to draw 2-, 3-and 4-colorable graphs in a Fary grid fashion in O(n 2)...

Let G = (V,E) be a graph. A k-coloring of a graph G is a labeling f : V (G) → T , where | T |= k and it is proper if the adjacent vertices have different labels. A graph is k-colorable if it has a proper k-coloring. The chromatic number χ(G) is the least k such that G is k-colorable. Here we study chromatic numbers in some kinds of Harary graphs. Mathematics Subject Classification: 05C15