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arXiv 2306.08069math.COcs.DM
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关于具有有界稀疏性参数的图的 $(n,m)$-色数

On $(n,m)$-chromatic numbers of graphs having bounded sparsity parameters

Sandip Das, Abhiruk Lahiri, Soumen Nandi, Sagnik Sen, S Taruni

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18 pages
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AI总结：

本文研究稀疏图与偏 2-树的 $(n,m)$-色数，建立其与荫度、无环色数及最大平均度的关系，并改进平面图和偏 2-树的色数界。

摘要：

一个 $(n,m)$-图的特征是具有 $n$ 种弧和 $m$ 种边。$(n,m)$-图 $G$ 到 $(n,m)$-图 $H$ 的同态是保持邻接关系、方向和类型的顶点映射。$G$ 的 $(n,m)$-色数记为 $χ_{n,m}(G)$，定义为使得存在从 $G$ 到 $H$ 的同态的 $|V(H)|$ 的最小值。$(n,m)$-图同态理论与和谐着色、无处零流等图论概念，以及二元谓词逻辑、Coxeter 群等其他数学主题存在联系，并且可应用于图数据库中的查询求值问题（QEP）。在本文中，我们证明 $G$ 的荫度受 $χ_{n,m}(G)$ 的某个函数界定，但反向并不成立。此外，我们证明 $G$ 的无环色数受 $χ_{n,m}(G)$ 的某个函数界定，而反向结果此前已经为人所知。进一步地，我们证明对于最大平均度小于 $2+\frac{2}{4(2n+m)-1}$ 的图族，包括围长至少为 $8(2n+m)$ 的平面图子族，其 $(n,m)$-色数等于 $2(2n+m)+1$。这改进了此前的发现：此前结果证明围长至少为 $10(2n+m)-4$ 的平面图的 $(n,m)$-色数为 $2(2n+m)+1$。文中确立了偏 2-树族 $\mathcal{T}_2$ 的 $(n,m)$-色数同时受 $(2n+m)$ 的二次函数给出的下界和上界约束，并且当 $(2n+m)=2$ 时下界是紧的。我们证明 $14\leq χ_{(0,3)}(\mathcal{T}_2)\leq15$ 以及 $14\leq χ_{(1,1)}(\mathcal{T}_2)\leq21$，这同时改进了已知下界和前一个上界。此外，就我们所知，我们为后一个上界提供了首个理论证明。

英文摘要：

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.

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