Polynomials associated with graph coloring and orientations

We study colorings and orientations of graphs in two related contexts. Firstly, we generalize Stanley's chromatic symmetric function using the k-balanced colorings of Pretzel to create a new graph invariant. We show that in fact this invariant is a quasisymmetric function which has a positive expansion in the fundamental basis. We also define a graph invariant generalizing the chromatic polynomial for which we prove some theorems analogous to well-known theorems about the chromatic polynomial. Secondly, we examine graphs and graph colorings in the context of the combinatorial Hopf algebras of Aguiar, Bergeron and Sottile. By doing so, we are able to obtain a new formula for the antipode of a Hopf algebra on graphs previously studied by Schmitt. We also obtain new interpretations of evaluations of the Tutte polynomial

On Colorings and Orientations of Signed Graphs

A classical theorem independently due to Gallai and Roy states that a graph G has a proper k-coloring if and only if G has an orientation without coherent paths of length k. An analogue of this result for signed graphs is proved in this article

On the Computational Complexity of Defining Sets

Suppose we have a family ${\cal F}$ of sets. For every $S \in {\cal F}$, a set $D \subseteq S$ is a {\sf defining set} for $({\cal F},S)$ if $S$ is the only element of $\cal{F}$ that contains $D$ as a subset. This concept has been studied in numerous cases, such as vertex colorings, perfect matchings, dominating sets, block designs, geodetics, orientations, and Latin squares. In this paper, first, we propose the concept of a defining set of a logical formula, and we prove that the computational complexity of such a problem is $\Sigma_2$-complete. We also show that the computational complexity of the following problem about the defining set of vertex colorings of graphs is $\Sigma_2$-complete: {\sc Instance:} A graph $G$ with a vertex coloring $c$ and an integer $k$. {\sc Question:} If ${\cal C}(G)$ be the set of all $\chi(G)$-colorings of $G$, then does $({\cal C}(G),c)$ have a defining set of size at most $k$? Moreover, we study the computational complexity of some other variants of this problem

On the phase transitions of graph coloring and independent sets

We study combinatorial indicators related to the characteristic phase transitions associated with coloring a graph optimally and finding a maximum independent set. In particular, we investigate the role of the acyclic orientations of the graph in the hardness of finding the graph's chromatic number and independence number. We provide empirical evidence that, along a sequence of increasingly denser random graphs, the fraction of acyclic orientations that are `shortest' peaks when the chromatic number increases, and that such maxima tend to coincide with locally easiest instances of the problem. Similar evidence is provided concerning the `widest' acyclic orientations and the independence number

A bivariate chromatic polynomial for signed graphs

We study Dohmen--P\"onitz--Tittmann's bivariate chromatic polynomial $c_\Gamma(k,l)$ which counts all $(k+l)$-colorings of a graph $\Gamma$ such that adjacent vertices get different colors if they are $\le k$. Our first contribution is an extension of $c_\Gamma(k,l)$ to signed graphs, for which we obtain an inclusion--exclusion formula and several special evaluations giving rise, e.g., to polynomials that encode balanced subgraphs. Our second goal is to derive combinatorial reciprocity theorems for $c_\Gamma(k,l)$ and its signed-graph analogues, reminiscent of Stanley's reciprocity theorem linking chromatic polynomials to acyclic orientations.Comment: 8 pages, 4 figure

Chromatic invariants of signed graphs

AbstractWe continue the study initiated in “Signed graph coloring” of the chromatic and Whitney polynomials of signed graphs. In this article we prove and apply to examples three types of general theorem which have no analogs for ordinary graph coloring. First is a balanced expansion theorem which reduces calculation of the chromatic and Whitney polynomials to that of the simpler balanced polynomials. Second is a group of formulas based on counting colorings by their magnitudes or their signs; among them are a combinatorial interpretation of signed coloring (which implies an equivalence between proper colorings of certain signed graphs and matching in ordinary graphs) and a signed-graphic switching formula (which for instance gives the polynomials of a two-graph in terms of those of its associated ordinary graphs). Third are addition/deletion formulas obtained by constructing one signed graph from another through adding and removing arcs; one such formula expresses the chromatic polynomial as a combination of those of ordinary graphs, while another (in one example) yields a complementation formula for ordinary matchings. The examples treated are the sign-symmetric graphs (among them in effect the classical root systems), all-negative graphs (corresponding to the even-cycle graphic matroid), signed complete graphs (equivalent to two-graphs), and two varieties of signed graphs associated with matchings and colorings of ordinary graphs. Our results are interpreted as counting the acyclic orientations of a signed graph; geometrically this means counting the faces of the corresponding arrangement of hyperplanes or zonotope