Line k-Arboricity in Product Networks

TL;DR

This paper investigates the linear k-arboricity of various graph products and specific network classes, providing bounds and exact values for these decompositions, which are useful in network design and analysis.

Contribution

It establishes general bounds for the linear k-arboricity of Cartesian, lexicographic, direct, and strong graph products, and computes values for specific network topologies.

Findings

Bounds for linear k-arboricity of Cartesian products.

Bounds for linear k-arboricity of other graph products.

Exact values for specific network topologies.

Abstract

A \emph{linear $k$-forest} is a forest whose components are paths of length at most $k$. The \emph{linear $k$-arboricity} of a graph $G$, denoted by ${\rm la}_k(G)$, is the least number of linear $k$-forests needed to decompose $G$. Recently, Zuo, He and Xue studied the exact values of the linear $(n-1)$-arboricity of Cartesian products of various combinations of complete graphs, cycles, complete multipartite graphs. In this paper, for general $k$ we show that $\max\{{\rm la}_{k}(G),{\rm la}_{\ell}(H)\}\leq {\rm la}_{\max\{k,\ell\}}(G\Box H)\leq {\rm la}_{k}(G)+{\rm la}_{\ell}(H)$ for any two graphs $G$ and $H$. Denote by $G\circ H$, $G\times H$ and $G\boxtimes H$ the lexicographic product, direct product and strong product of two graphs $G$ and $H$, respectively. We also derive upper and lower bounds of ${\rm la}_{k}(G\circ H)$, ${\rm la}_{k}(G\times H)$ and ${\rm la}_{k}(G\boxtimes H)$ in this paper. The linear $k$-arboricity of a $2$-dimensional grid graph, a $r$-dimensional mesh, a $r$-dimensional torus, a $r$-dimensional generalized hypercube and a $2$-dimensional hyper Petersen network are also studied.