A graph is called to be uniquely list colorable, if it admits a list assignment which induces a unique list coloring. We study uniquely list colorable graphs with a restriction on the number of colors used. In this way we generalize a theorem which characterizes uniquely 2–list colorable graphs. We introduce the uniquely list chromatic number of a graph and make a conjecture about it which is a...

We address a question posed by T. Sibley and Stan Wagon. They proved that rhombic Penrose tilings in the plane can be 3-colored, but a key lemma of their proof fails in the natural 3D generalization. In that generalization, an object is built from bricks, each of which is a parellelopiped, and they are glued face-toface. The question is: How many colors are needed to color the bricks of any suc...

A graph G is called (k, 1)-colorable, if the vertex set of G can be partitioned into subsets V1 and V2 such that the graph G[V1] induced by the vertices of V1 has maximum degree at most k and the graph G[V2] induced by the vertices of V2 has maximum degree at most 1. We prove that every graph with a maximum average degree less than 10k+22 3k+9 admits a (k, 1)-coloring, where k ≥ 2. In particula...

Graph coloring for 3-colorable graphs receives very much attention by many researchers in theoretical computer science. Deciding 3-colorability of a graph is a well-known NP-complete problem. So far, the best known polynomial approximation algorithm achieves a factor of O(n0.2072), and there is a strong evidence that there would be no polynomial time algorithm to color 3-colorable graphs using ...

DP-coloring is a generalization of list-coloring, which was introduced by Dvořák and Postle. Zhang showed that every planar graph with neither adjacent triangles nor 5-, 6-, 9-cycles 3-choosable. Liu et al. without 4-, 6- DP-3-colorable. In this paper, we show DP-3-colorable, generalizes these results. Yu gave three Bordeaux-type results showing that: (i) the distance at least no 5-cycles DP-3-...

A graph is (k; d)-colorable if one can color the vertices with k colors such that no vertex is adjacent to more than d vertices of the same color. In this paper we investigate the existence of such colorings in surfaces. It is shown that a toroidal graph is (3; 2)and (5; 1)colorable, and that a graph of genus is ( =(d + 1) + 4; d)-colorable, where is the maximum chromatic number of a graph embe...

A graph G = (V,E) is r-equitably k-colorable if there exists a partition of V into k independent sets V1, V2, · · · , Vk such that | |Vi| − |Vj | | ≤ r for all i, j ∈ {1, 2, · · · , k}. In this note, we show that if two trees T1 and T2 of order at least two are r-equitably k-colorable for r ≥ 1 and k ≥ 3, then all trees obtained by adding an arbitrary edge between T1 and T2 are also r-equitably...

This lecture deals with the problem of proper vertex colorings of graphs. More speciically, we are interested in coloring a given 3-colorable graph with as few colors as we can. It is easy to see that it is NP-Hard to nd a 3 coloring for any given 3-colorable graph-this would enable deciding whether a general graph is 3-colorable. If a polynomial-time algorithm existed for 3-coloring a 3-colora...

An injective coloring of a graph is a vertex coloring where two vertices have distinct colors if a path of length two exists between them. In this paper some results on injective colorings of planar graphs with few colors are presented. We show that all planar graphs of girth ≥19 and maximum degree ∆ are injectively ∆-colorable. We also show that all planar graphs of girth ≥10 are injectively (...