On (n,m)-chromatic numbers of graphs having bounded sparsity parameters
Abstract
An (n,m)-graph is characterised by having n types of arcs and m types of edges. A homomorphism of an (n,m)-graph G to an (n,m)-graph H, is a vertex mapping that preserves adjacency, direction, and type. The (n,m)-chromatic number of G, denoted by n,m(G), is the minimum value of |V(H)| such that there exists a homomorphism of G to H. The theory of homomorphisms of (n,m)-graphs have connections with graph theoretic concepts like harmonious coloring, nowhere-zero flows; with other mathematical topics like binary predicate logic, Coxeter groups; and has application to the Query Evaluation Problem (QEP) in graph database. In this article, we show that the arboricity of G is bounded by a function of n,m(G) but not the other way around. Additionally, we show that the acyclic chromatic number of G is bounded by a function of n,m(G), a result already known in the reverse direction. Furthermore, we prove that the (n,m)-chromatic number for the family of graphs with a maximum average degree less than 2+ 24(2n+m)-1, including the subfamily of planar graphs with girth at least 8(2n+m), equals 2(2n+m)+1. This improves upon previous findings, which proved the (n,m)-chromatic number for planar graphs with girth at least 10(2n+m)-4 is 2(2n+m)+1. It is established that the (n,m)-chromatic number for the family T2 of partial 2-trees is both bounded below and above by quadratic functions of (2n+m), with the lower bound being tight when (2n+m)=2. We prove 14 ≤ (0,3)(T2) ≤ 15 and 14 ≤ (1,1)(T2) ≤ 21 which improves both known lower bounds and the former upper bound. Moreover, for the latter upper bound, to the best of our knowledge we provide the first theoretical proof.