The complexity of counting edge colorings and a dichotomy for some higher domain Holant problems

We show that an effective version of Siegel’s Theorem on finiteness of integer solutions and an application of elementary Galois theory are key ingredients in a complexity classification of some Holant problems. These Holant problems, denoted by Holant(f), are defined by a symmetric ternary function f that is invariant under any permutation of the κ ≥ 3 domain elements. We prove that Holant(f) exhibits a complexity dichotomy. This dichotomy holds even when restricted to planar graphs. A special case of this result is that counting edge κ-colorings is #P-hard over planar 3-regular graphs for κ ≥ 3. In fact, we prove that counting edge κ-colorings is #P-hard over planar r-regular graphs for all κ ≥ r ≥ 3. The problem is polynomial-time computable in all other parameter settings. The proof of the dichotomy theorem for Holant(f) depends on the fact that a specific polynomial p(x, y) has an explicitly listed finite set of integer solutions, and the determination of the Galois groups of some specific polynomials. In the process, we also encounter the Tutte polynomial, medial graphs, Eulerian partitions, Puiseux series, and a certain lattice condition on the (logarithm of) the roots of polynomials.

Orientations of Graphs with Prescribed Weighted Out-Degrees

If we want to apply Galvin's kernel method to show that a graph G satisfies a certain coloring property, we have to find an appropriate orientation of G. This motivated us to investigate the complexity of the following orientation problem. The input is a graph G and two vertex functions {Mathematical expression}. Then the question is whether there exists an orientation D of G such that each vertex {Mathematical expression} satisfies {Mathematical expression}. On one hand, this problem can be solved in polynomial time if g(v) = 1 for every vertex {Mathematical expression}. On the other hand, as proved in this paper, the problem is NP-complete even if we restrict it to graphs which are bipartite, planar and of maximum degree at most 3 and to functions f, g where the permitted values are 1 and 2, only. We also show that the analogous problem, where we replace g by an edge function {Mathematical expression} and where we ask for an orientation D such that each vertex {Mathematical expression} satisfies {Mathematical expression}, is NP-complete, too. Furthermore, we prove some new results related to the (f, g)-choosability problem, or in our terminology, to the list-coloring problem of weighted graphs. In particular, we use Galvin's theorem to prove a generalization of Brooks's theorem for weighted graphs. We show that if a connected graph G has a block which is neither a complete graph nor an odd cycle, then G has a kernel perfect super-orientation D such that {Mathematical expression} for every vertex {Mathematical expression}. © 2013 Springer Japan

On list critical graphs

AbstractIn this paper we discuss some basic properties of k-list critical graphs. A graph G is k-list critical if there exists a list assignment L for G with |L(v)|=k−1 for all vertices v of G such that every proper subgraph of G is L-colorable, but G itself is not L-colorable. This generalizes the usual definition of a k-chromatic critical graph, where L(v)={1,…,k−1} for all vertices v of G. While the investigation of k-critical graphs is a well established part of coloring theory, not much is known about k-list critical graphs. Several unexpected phenomena occur, for instance a k-list critical graph may contain another one as a proper induced subgraph, with the same value of k. We also show that, for all 2≤p≤k, there is a minimal k-list critical graph with chromatic number p. Furthermore, we discuss the question, for which values of k and n is the complete graph Knk-list critical. While this is the case for all 5≤k≤n, Kn is not 4-list critical if n is large

A New Lower Bound on the Number of Edges in Colour-Critical Graphs

A graph G is called k-critical if it has chromatic number k, but every proper subgraph of it is (k \Gamma 1)--colourable. We prove that every k--critical graph (k 6) on n k + 2 vertices has at least 1 2 (k \Gamma 1+ k\Gamma3 (k\Gammac)(k\Gamma1)+k\Gamma3 )n edges where c = (k \Gamma 5)( 1 2 \Gamma 1 (k\Gamma1)(k\Gamma2) ). This improves earlier bounds established by Gallai [9] and, more recently, by Krivelevich [17]. 1 Introduction A graph G is k--critical for some integer k 1 if G is not (k \Gamma1)--colourable but every proper subgraph of G is (k \Gamma 1)--colourable. Then every k--critical graph has chromatic number k and every k--chromatic graph contains a k--critical subgraph. The importance of the concept of criticality consists in the fact that problems for k--chromatic graphs may often be reduced to problems for k--critical graphs, and that the class of k--critical graphs is a narrow subclass of the class of k--chromatic graphs. Critical graphs were first defined a..

Colour-critical graphs with few edges

A graph G is called k-critical if G is k-chromatic but every proper subgraph of G has chromatic number at most k- 1. In this paper the following result is proved. If G is a k-critical graph (k>~4) on n vertices, then 21E(G)I>(k- 1)n ÷ ((k- 3)/(k 2- 3))n + k- 4 where n>~k +