LINEAR 2-ARBORICITY OF THE COMPLETE GRAPH

Abstract

A linear $k$-forest is a graph whose components are paths with lengths at most $k$. The minimum number of linear $k$-forests needed to decompose a graph $G$ is the linear $k$-arboricity of $G$ and denoted by $la_{k}(G)$. In this paper, we settle the cases left in determining the linear $2$-arboricity of the complete graph $K_n$. Mainly, we prove that $la_{2} (K_{12t+10}) = la_{2}(K_{12t+11}) = 9t+8$ for any $t \geq 0$.