Pavol Hell

On the complexity of H -coloring

Abstract Let H be a fixed graph, whose vertices are referred to as ‘colors’. An H-coloring of a graph G is an assignment of ‘colors’ to the vertices of G such that adjacent vertices of G obtain adjacent ‘colors’. (An H-coloring of G is just a homomorphism G → H). The following H-coloring problem has been the object of recent interest: Instance: A graph G. Question: Is it possible to H-color the graph G? H-colorings generalize traditional graph colorings, and are of interest in the study of grammar interpretations. Several authors have studied the complexity of the H-coloring problem for various (families of) fixed graphs H. Since there is an easy H-colorability test when H is bipartite, and since all other examples of the H-colorability problem that were treated (complete graphs, odd cycles, complements of odd cycles, Kneser graphs, etc.) turned out to be NP-complete, the natural conjecture, formulated in several sources (including David Johnsons NP-completeness column), asserts that the H-coloring problem is NP-complete for any non-bipartite graph H. We give a proof of this conjecture.

On the History of the Minimum Spanning Tree Problem

It is standard practice among authors discussing the minimum spanning tree problem to refer to the work of Kruskal(1956) and Prim (1957) as the sources of the problem and its first efficient solutions, despite the citation by both of Boruvka (1926) as a predecessor. In fact, there are several apparently independent sources and algorithmic solutions of the problem. They have appeared in Czechoslovakia, France, and Poland, going back to the beginning of this century. We shall explore and compare these works and their motivations, and relate them to the most recent advances on the minimum spanning tree problem.

The core of a graph

Abstract The core of a graph is its smallest subgraph which also is a homomorphic image. It turns out the core of a finite graph is unique (up to isomorphism) and is also its smallest retract. We investigate some homomorphism properties of cores and conclude that it is NP-complete to decide whether or not a graph is its own core. (A similar conclusion is reached about testing whether or not a graph is rigid, i.e., admits a non-identity homomorphism to itself.) We also give a polynomial-time verifiable condition for a graph of small independence number to be its own core.

List Homomorphisms and Circular Arc Graphs

G, H, and lists , a list homomorphism of G to Hwith respect to the listsL is a mapping , such that for all , and for all . The list homomorphism problem for a fixed graph H asks whether or not an input graph G together with lists , , admits a list homomorphism with respect to L. We have introduced the list homomorphism problem in an earlier paper, and proved there that for reflexive graphs H (that is, for graphs H in which every vertex has a loop), the problem is polynomial time solvable if H is an interval graph, and is NP-complete otherwise. Here we consider graphs H without loops, and find that the problem is closely related to circular arc graphs. We show that the list homomorphism problem is polynomial time solvable if the complement of H is a circular arc graph of clique covering number two, and is NP-complete otherwise. For the purposes of the proof we give a new characterization of circular arc graphs of clique covering number two, by the absence of a structure analogous to Gallais asteroids. Both results point to a surprising similarity between interval graphs and the complements of circular arc graphs of clique covering number two.

Linear-Time Representation Algorithms for Proper Circular-Arc Graphs and Proper Interval Graphs

Our main result is a linear-time (that is, time

A note on the star chromatic number

O(m+n)

On the Complexity of General Graph Factor Problems

) algorithm to recognize and represent proper circular-arc graphs. The best previous algorithm, due to A. Tucker, has time complexity

On the completeness of a generalized matching problem

O(n^2)

List Homomorphisms to Reflexive Graphs

. We take advantage of the fact that (among connected graphs) proper circular-arc graphs are precisely the graphs orientable as local tournaments, and we use a new characterization of local tournaments. The algorithm depends on repeated representation of portions of the input graph as proper interval graphs. Thus we also find it useful to give a new linear-time algorithm to represent proper interval graphs. This latter algorithm also depends on an orientation characterization of proper interval graphs. It is conceptually simple and does not use complex data structures. As a byproduct of the correctness proof of the algorithm, we also obtain a new proof of a characterization of proper interval graphs by forbidden subgraphs.

Duality and Polynomial Testing of Tree Homomorphisms

A. Vince introduced a natural generalization of graph coloring and proved some basic facts, revealing it to be a concept of interest. His work relies on continuous methods. In this note we make some simple observations that lead to a purely combinatorial treatment. Our methods yield shorter proofs and offer further insight.