Loose Edge-Connection of Graphs

Abstract

AbstractIn 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 $${{\,\textrm{lec}\,}}(G)$$ 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 $${{\,\textrm{lec}\,}}(G) = 2$$ lec ( G ) = 2 for graphs G of diameter 2, is an NP-complete problem. Furthermore, we characterize all complete bipartite graphs $$K_{r,s}$$ K r , s with $${{\,\textrm{lec}\,}}(K_{r,s}) = 2$$ lec ( K r , s ) = 2 .