Loose edge-connection of graphs

TL;DR

This paper introduces the concept of loose edge-connection in graphs, determines its value for graphs with diameter at least 3, and explores computational complexity and specific graph classes related to this parameter.

Contribution

It defines the loose edge-connection number, computes it for certain graphs, and proves NP-completeness for deciding the parameter in diameter-2 graphs.

Findings

Exact values of loose edge-connection number for graphs with diameter ≥ 3

NP-completeness of deciding if the number equals 2 for diameter-2 graphs

Characterization of complete bipartite graphs with loose edge-connection number 2

Abstract

In the last years, connection concepts such as rainbow connection and proper connection appeared in graph theory and obtained a lot of attention. In this paper, we investigate the loose edge-connection of graphs. A connected edge-coloured graph $G$ is loose edge-connected if between any two of its vertices there is a path of length one, or a bi-coloured path of length two, or a path of length at least three with at least three colours used on its edges. The minimum number of colours, used in a loose edge-colouring of $G$, is called the loose edge-connection number and denoted $\lec(G)$. We determine the precise value of this parameter for any simple graph $G$ of diameter at least 3. We show that deciding, whether $\lec(G) = 2$ for graphs $G$ of diameter 2, is an NP-complete problem. Furthermore, we characterize all complete bipartite graphs $K_{r,s}$ with $\lec(K_{r,s}) = 2$.