On $(n,m)$-chromatic numbers of graphs having bounded sparsity parameters
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2023
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| author | Das, Sandip Lahiri, Abhiruk Nandi, Soumen Sen, Sagnik Taruni, S |
| author_facet | Das, Sandip Lahiri, Abhiruk Nandi, Soumen Sen, Sagnik Taruni, S |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_08069 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On $(n,m)$-chromatic numbers of graphs having bounded sparsity parameters Das, Sandip Lahiri, Abhiruk Nandi, Soumen Sen, Sagnik Taruni, S Combinatorics Discrete Mathematics 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. |
| title | On $(n,m)$-chromatic numbers of graphs having bounded sparsity parameters |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2306.08069 |