Graphs with multiplicative vertex-coloring 2-edge-weightings

Abstract

A k-weighting w of a graph is an assignment of an integer weight w(e)in {1,...k} to each edge e. Such an edge weighting induces a vertex coloring c defined by c(v)=mathop {displaystyle {prod }}limits _{vin e}w(e). A k-weighting of a graph G is multiplicative vertex-coloring if the induced coloring c is proper, i.e., c(u)ne c(v) for any edge uvin E(G). This paper studies the parameter mu (G), which is the minimum k for which G has a multiplicative vertex-coloring k-weighting. Chang, Lu, Wu, Yu investigated graphs with additive vertex-coloring 2-weightings (they considered sums instead of products of incident edge weights). In particular, they proved that 3-connected bipartite graphs, bipartite graphs with the minimum degree 1, and r-regular bipartite graphs with rge 3 permit an additive vertex-coloring 2-weighting. In this paper, the multiplicative version of the problem is considered. It was shown in Skowronek-Kaziów (Inf Process Lett 112:191–194, 2012) that mu (G)le 4 for every graph G. It was also proved that every 3-colorable graph admits a multiplicative vertex-coloring 3 -weighting. A natural problem to consider is whether every 2-colorable graph (i.e., a bipartite graph) has a multiplicative vertex-coloring 2-weighting. But the answer is no, since the cycle C_{6} and the path P_{6} do not admit a multiplicative vertex-coloring 2-weighting. The paper presents several classes of 2-colorable graphs for which mu (G)=2, including trees with at least two adjacent leaf edges, bipartite graphs with the minimum degree 3 and bipartite graphs G=(A,B,E) with even left| Aright| or left| Bright| .