2020/09/11 by Christopher Duffy, Duffy, Christopher, Fabien Jacques +5
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2009.05439
openalex publication_date 2020/09/11 · openalex created_date 2023/06/28 · openalex updated_date 2026/07/28
A 2-edge-colored graph or a signed graph is a simple graph with two types of edges. A homomorphism from a 2-edge-colored graph G to a 2-edge-colored graph H is a mapping φ: V(G) → V(H) that maps every edge in G to an edge of the same type in H. Switching a vertex v of a 2-edge-colored or signed graph corresponds to changing the type of each edge incident to v. There is a homomorphism from the signed graph G to the signed graph H if after switching some subset of the vertices of G there is a 2-edge-colored homomorphism from G to H. The chromatic number of a 2-edge-colored (resp. signed) graph G is the order of a smallest 2-edge-colored (resp. signed) graph H such that there is a homomorphism from G to H. The chromatic number of a class of graph is the maximum of the chromatic numbers of the graphs in the class. We study the chromatic numbers of 2-edge-colored and signed graphs (connected and not necessarily connected) of a given bounded maximum degree. More precisely, we provide exact bounds for graphs of maximum degree 2. We then propose specific lower and upper bounds for graphs of maximum degree 3, 4, and 5. We finally propose general bounds for graphs of maximum degree k, for every k.