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20.10.2015 | Ausgabe 1/2017 Open Access

Journal of Combinatorial Optimization 1/2017

Graphs with multiplicative vertex-coloring 2-edge-weightings

Zeitschrift:
Journal of Combinatorial Optimization > Ausgabe 1/2017
Autor:
Joanna Skowronek-Kaziów

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 \(c(v)=\mathop {\displaystyle {\prod }}\limits _{v\in e}w(e).\) 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 \(uv\in E(G).\) 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 \(r\ge 3\) 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 \(\left| A\right| \) or \(\left| B\right| \).

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