Michael O. Albertson

Homomorphisms of 3-chromatic graphs

Abstract This paper examines the effect of a graph homomorphism upon the chromatic difference sequence of a graph. Our principal result (Theorem 2) provides necessary conditions for the existence of a homomorphism onto a prescribed target. As a consequence we note that iterated cartesian products of the Petersen graph form an infinite family of vertex transitive graphs no one of which is the homomorphic image of any other. We also prove that there is a unique minimal element in the homomorphism order of 3-chromatic graphs with non-monotonic chromatic difference sequences (Theorem 1). We include a brief guide to some recent papers on graph homomorphisms.

Every planar graph has an acyclic 7-coloring

A proper coloring of the vertices of a graph is said to be acyclic provided that no cycle is two colored. We prove that every planar graph has an acyclic seven coloring.

The subchromatic number of a graph

The subchromatic number X S ( G ) of a graph G = ( V, E ) is the smallest order k of a partition {V 1 , V 2 , …, V k } of the vertices V ( G ) such that the subgraph ( V i ) induced by each subset V i consists of a disjoint union of complete subgraphs. By definition, X s ( G ) ⩽ ( G ), the chromatic number of G . This paper develops properties of this lower bound for the chromatic number.

You Can't Paint Yourself into a Corner

Thomassen posed the following problem: SupposeGis a planar graph andW?V(G) such that the distance between any two vertices inWis at least 100. Can a 5-coloring ofWbe extended to a 5-coloring ofG? It is known that no 4-coloring result of this nature can hold. We provide a best possible solution to Thomassens problem as well as some generalizations.

Extending Graph Colorings

Suppose ?(G)=r and P?V(G). It is known that if the distance between any two vertices in P is at least 4, then any (r+1)-coloring of P extends to an (r+1)-coloring of all of G, but an r-coloring of P might not extend to an r-coloring of G. We show that if the distance between any two vertices in P is at least 3, then an (r+1)-coloring of P can be extended to a ?(3r+1)/2?-coloring of G. Kostochka showed that if P induces a set of k-cliques whose pairwise distance is at least 4k, then an (r+1)-coloring of P can be extended to an (r+1)-coloring of G. We give Kostochkas proof and more precise results concerning the distance required between precolored components. For example, we show that when k=r, there is a coloring extension provided the cliques have pairwise distance at least 3k. We relate the structure of the precolored components to the number of extra colors needed in a coloring extension theorem. We construct families of graphs to show that all of the above results are close to being best possible.

On the independence ratio of a graph

This paper presents some recent results on lower bounds for independence ratios of graphs of positive genus and shows that in a limiting sense these graphs have the same independence ratios as do planar graphs. This last result is obtained by an application of Mengers Theorem to show that every triangulation of a surface of positive genus has a short cycle which does not separate the graph and is non-contractible on that surface.

Duality and perfection for edges in cliques

Abstract Given a graph G , let K ( G ) denote the graph whose vertices correspond with the edges of G . Two vertices of K ( G ) are joined by an edge if the corresponding edges in G are contained in a clique. This paper investigates some properties of G which force duality theorems for K ( G ).

The chromatic difference sequence of a graph

Abstract Let α k ( G ) denote the maximum number of vertices in a k -colorable subgraph of G . Set α k (G)= α k ( G )− α ( k −1) ( G ). The sequence a 1 ( G ), a 2 ( G ),… is called the chromatic difference sequence of the graph G . We present necessary and sufficient conditions for a sequence to be the chromatic difference sequence of some 4-colorable graph.

A lower bound for the independence number of a planar graph

A subset of the vertices of a graph is independent if no two vertices in the subset are adjacent. The independence number α(G) is the maximum number of vertices in an independent set. We prove that if G is a planar graph on N vertices then α(G)/N ⩾ 29.

Partial list colorings

Abstract Suppose G is an s -choosable graph with n vertices, and every vertex of G is assigned a list of t colors. We conjecture that at least (t/s)n of the vertices of G can be colored from these lists. We provide lower bounds and consider related questions. For instance, we show that if G is χ -colorable (rather than being s -choosable), then more than (1−((χ−1)/χ) t )n of the vertices of G can be colored from the lists and that this is asymptotically best possible. We include a number of open questions.