On the achromatic index of Johnson graphs $J(n,2)$

Abstract

In this paper, we study proper and complete edge-colorings of Johnson graphs $J(n,2)$, also called $n$-triangular graphs. They are isomorphic both to the 2-token graphs of complete graphs and to the line graphs of complete graphs. A $t$-edge-coloring of a graph $G$ is a function that assigns one color from $\{1,2,\ldots,t\}$ to each edge. Such a coloring is called proper if no two incident edges receive the same color, and complete if every pair of distinct colors appears on a pair of incident edges. The achromatic index, denoted by $\alpha_2(G)$, is the largest integer $t$ for which $G$ admits a proper and complete $t$-edge-coloring. We establish new lower and upper bounds for $\alpha_2(J(n,2))$, provide explicit proper and complete edge-colorings attaining the lower bounds, and determine the exact value of $\alpha_2(J(n,2))$ for several values of $n$.