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

Nowhere-Zero 3-Flows in Signed Graphs

Tutte observed that every nowhere-zero $k$-flow on a plane graph gives rise to a $k$-vertex-coloring of its dual, and vice versa. Thus nowhere-zero integer flow and graph coloring can be viewed as dual concepts. Jaeger further shows that if a graph $G$ has a face-$k$-colorable 2-cell embedding in some orientable surface, then it has a nowhere-zero $k$-flow. However, if the surface is nonorientable, then a face-$k$-coloring corresponds to a nowhere-zero $k$-flow in a signed graph arising from $G$. Graphs embedded in orientable surfaces are therefore a special case that the corresponding signs are all positive. In this paper, we prove that if an 8-edge-connected signed graph admits a nowhere-zero integer flow, then it has a nowhere-zero 3-flow. Our result extends Thomassen\u27s 3-flow theorem on 8-edge-connected graphs to the family of all 8-edge-connected signed graphs. And it also improves Zhu\u27s 3-flow theorem on 11-edge-connected signed graphs

Coloring permutation-gain graphs

Correspondence colorings of graphs were introduced in 2018 by Dvořák and Postle as a generalization of list colorings of graphs which generalizes ordinary graph coloring. Kim and Ozeki observed that correspondence colorings generalize various notions of signed-graph colorings which again generalizes ordinary graph colorings. In this note we state how correspondence colorings generalize Zaslavsky's notion of gain-graph colorings and then formulate a new coloring theory of permutation-gain graphs that sits between gain-graph coloring and correspondence colorings. Like Zaslavsky's gain-graph coloring, our new notion of coloring permutation-gain graphs has well defined chromatic polynomials and lifts to colorings of the regular covering graph of a permutation-gain graph

Online list coloring for signed graphs

We generalize the notion of online list coloring to signed graphs. We define the online list chromatic number of a signed graph, and prove a generalization of Brooks’ Theorem. We also give necessary and sufficient conditions for a signed graph to be degree paintable, or degree choosable. Finally, we classify the 2-list-colorable and 2-list-paintable signed graphs