Conflict-free connection numbers of line graphs

TL;DR

This paper studies the conflict-free connection numbers of line graphs and claw-free graphs, establishing bounds, exact values, and relationships between iterated line graphs and their conflict-free connectivity.

Contribution

It provides new bounds and exact values for conflict-free connection numbers of line graphs and claw-free graphs, and explores how these numbers change with iterated line graphs.

Findings

For any connected graph G, a positive k exists such that cfc(L^k(G)) ≤ 2.

Exact cfc values are obtained for connected claw-free and line graphs.

For most graphs, cfc(L^{k+1}(G)) ≤ cfc(L^k(G)), except for stars of order ≥ 5 when k=1.

Abstract

A path in an edge-colored graph is called \emph{conflict-free} if it contains at least one color used on exactly one of its edges. An edge-colored graph $G$ is \emph{conflict-free connected} if for any two distinct vertices of $G$, there is a conflict-free path connecting them. For a connected graph $G$, the \emph{conflict-free connection number} of $G$, denoted by $cfc(G)$, is defined as the minimum number of colors that are required to make $G$ conflict-free connected. In this paper, we investigate the conflict-free connection numbers of connected claw-free graphs, especially line graphs. We first show that for an arbitrary connected graph $G$, there exists a positive integer $k$ such that $cfc(L^k(G))\leq 2$. Secondly, we get the exact value of the conflict-free connection number of a connected claw-free graph, especially a connected line graph. Thirdly, we prove that for an arbitrary connected graph $G$ and an arbitrary positive integer $k$, we always have $cfc(L^{k+1}(G))\leq cfc(L^k(G))$, with only the exception that $G$ is isomorphic to a star of order at least~$5$ and $k=1$. Finally, we obtain the exact values of $cfc(L^k(G))$, and use them as an efficient tool to get the smallest nonnegative integer $k_0$ such that $cfc(L^{k_0}(G))=2$.