On $(n,m)$-chromatic numbers of graphs having bounded sparsity parameters
Sandip Das, Abhiruk Lahiri, Soumen Nandi, Sagnik Sen, S Taruni
TL;DR
This paper studies the (n,m)-chromatic number χ_{n,m}(G) of colored mixed graphs and its relation to graph sparsity parameters. It shows that arb(und(G)) is bounded by a function of χ_{n,m}(G) while the reverse need not hold, and that the acyclic chromatic number χ_a(G) is also bounded by χ_{n,m}(G). It derives an exact bound χ_{n,m}(G) = 2(2n+m)+1 for graphs with mad(G) below a critical threshold, with planar graphs of girth at least 8(2n+m) achieving this bound, and strengthens results for planar graphs via corollaries on circular chromatic numbers. For partial 2-trees, it tightens the known bounds to 14 ≤ χ_{0,3}(T_2) ≤ 15 and 14 ≤ χ_{1,1}(T_2) ≤ 21, providing explicit minimal universal-bound constructions and the first theoretical proof for the latter upper bound. These results advance the understanding of (n,m)-chromatic numbers in sparse graph families and raise questions about broader connections to other sparsity notions.
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 $χ_{n,m}(G)$, 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 $χ_{n,m}(G)$ but not the other way around. Additionally, we show that the acyclic chromatic number of $G$ is bounded by a function of $χ_{n,m}(G)$, 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 $2+ \frac{2}{4(2n+m)-1}$, 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 $\mathcal{T}_2$ 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 $14 \leq χ_{(0,3)}(\mathcal{T}_2) \leq 15$ and $14 \leq χ_{(1,1)}(\mathcal{T}_2) \leq 21$ 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.