In this paper, a combination of algorithms for graph coloring is discussed. The algorithms of KargerMotwani-Sudan and Blum-Karger can be combined in a simple way to yield a polynomial-time algorithm for an Õ(n7/18)-coloring of any n-vertex 4-colorable graph. This result can be generalized to k-colorable graphs to obtain a coloring with Õ(n 1− 1 (k+1)/3−4/(11k2−11k) ) colors, which slightly impr...

Let m∗(n) be the minimum number of edges in an n-uniform simple hypergraph that is not two colorable. We prove that m∗(n) = Ω(4n/ ln(n)). Our result generalizes to r-coloring of b-simple uniform hypergraphs. For fixed r and b we prove that a maximum vertex degree in b-simple n-uniform hypergraph that is not r-colorable must be Ω(rn/ ln(n)). By trimming arguments it implies that every such graph...

In the course of extending Grötzsch’s theorem, we prove that every triangle-free graph without a K5-minor is 3-colorable. It has been recently proved that every triangle-free planar graph admits a homomorphism to the Clebsch graph. We also extend this result to the class of triangle-free graphs without a K5-minor. This is related to some conjectures which generalize the Four-Color Theorem. Whil...

Suppose G is a graph embedded in S g with width (also known as edge width) at least 264(2 g ?1). If P V (G) is such that the distance between any two vertices in P is at least 16, then any 5-coloring of P extends to a 5-coloring of all of G. We present similar extension theorems for 6-and 7-chromatic toroidal graphs, for 3-colorable large-width graphs embedded on S g with every face even sided,...

Is every zonohedron 3-colorable when viewed as a planar map? This question arose out of work described in [RSW01]. An equivalent question, under a different guise, is posed in [FHNS00]: Is the arrangement graph of great circles on the sphere 3colorable? Assume no three circles meet at a point, so that this graph is 4-regular. Circle graphs in the plane can require four colors [Koe90], so the ke...

A defining set (of vertex coloring) of a graph G is a set of vertices S with an assignment of colors to its elements which has a unique completion to a proper coloring of G. We define a minimal defining set to be a defining set which does not properly contain another defining set. If G is a uniquely vertex colorable graph, clearly its minimum defining sets are of size χ(G) − 1. It is shown that...

A graph G is strongly set colorable if V (G) ∪ E(G) can be assigned distinct nonempty subsets of a set of order n, where |V (G)| + |E(G)| = 2n − 1, such that each edge is assigned the symmetric difference of its end vertices. The principal result is that the path P2n−1 is strongly set colorable for n ≥ 5, disproving a conjecture of S.M. Hegde. We also prove another conjecture of S.M. Hegde on a...

We consider the problem of clique coloring, that is, coloring the vertices of a given graph such that no (maximal) clique of size at least two is monocolored. It is known that interval graphs are 2-clique colorable. In this paper we prove that B1-EPG graphs (edge intersection graphs of paths on a grid, where each path has at most one bend) are 4-clique colorable. Moreover, given a B1-EPG repres...

A graph G is equitably k-choosable if for any k-uniform list assignment L, G is L-colorable and each color appears on at most d|V (G)|/ke vertices. A graph G is equitable kcolorable if G has a proper vertex coloring with k colors such that the size of the color classes differ by at most 1. In this paper, we prove that if G is a planar graph without 5and 7-cycles, then G is equitably k-choosable...

We introduce the idea of an n-simplex graph and games upon simplicial complexes. We then define moves on a labeled graph and pose the problem of whether given two labelings of a graph it is possible to change one into another via these moves. We then solve the problem for a given class of graphs. Once having found a solution for a given class of graphs we determine the number of different solut...