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 {{,textrm{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 {{,textrm{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 {{,textrm{lec},}}(K_{r,s}) = 2.
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