Graphs with multiplicative vertex-coloring 2-edge-weightings

A k-weighting w of a graph is an assignment of an integer weight $$w(e)\in \{1,...k\}$$ w ( e ) ∈ { 1 , . . . k } to each edge e. Such an edge weighting induces a vertex coloring c defined by $$c(v)=\mathop {\displaystyle {\prod }}\limits _{v\in e}w(e).$$ c ( v ) = ∏ v ∈ e w ( e ) . A k-weighting of a graph G is multiplicative vertex-coloring if the induced coloring c is proper, i.e., $$c(u)\ne c(v)$$ c ( u ) ≠ c ( v ) for any edge $$uv\in E(G).$$ u v ∈ E ( G ) . This paper studies the parameter $$\mu (G),$$ μ ( 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 $$r\ge 3$$ r ≥ 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 $$\mu (G)\le 4$$ μ ( G ) ≤ 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 $$C_{6}$$ C 6 and the path $$P_{6}$$ P 6 do not admit a multiplicative vertex-coloring 2-weighting. The paper presents several classes of 2-colorable graphs for which $$\mu (G)=2$$ μ ( G ) = 2 , including trees with at least two adjacent leaf edges, bipartite graphs with the minimum degree 3 and bipartite graphs $$G=(A,B,E)$$ G = ( A , B , E ) with even $$\left| A\right| $$ A or $$\left| B\right| $$ B .