Vertex-coloring Edge-weightings of Graphs

Abstract

A k-edge-weighting of a graph G is a mapping w : E(G) → {1, 2, . . ., k}. An edge-weighting w induces a vertex coloringfor any edge uv. The current paper studies the parameter µ(G), which is the minimum k for which G has a vertex-coloring k-edgeweighting. Exact values of µ(G) are determined for several classes of graphs, including trees and r-regular bipartite graph with r ≥ 3.

Graphs with multiplicative vertex-coloring 2-edge-weightings

“…Different versions of vertex coloring from an edge k-weighting (by considering the sums, products, sequences, sets, or multisets of incident edge weights) were investigated by many authors in Addario-Berry et al (2005), Bartnicki et al (2009), Chang et al (2011, Kalkowski et al (2010), Karoński et al (2004), Lu et al (2011), Skowronek-Kaziów (2012, Stevens and Seamone (2013). Some authors distinguish all the vertices in a graph by their product colors (product irregularity strength of graphs, see Anholcer (2009Anholcer ( , 2014, Darda and Hujdurovic (2014)).…”

“…In 2011, Chang, Lu, Wu, Yu, and Zhang [Chang et al (2011] considered the graphs with additive vertex-coloring 2-weightings. In particular, they proved that 3-connected bipartite graphs, bipartite graphs with the minimum degree 1, and r -regular bipartite graphs with r ≥ 3 permit an additive vertex-coloring 2-weighting.…”

Derived colourings of graphs

“…In Chapter 3, we will establish a variety of classes of graphs for which this bound holds. In many cases the results given provide generalizations of results given in [28], in that they hold for weighting sets of size two other than {1,2}. We will also prove that there is a set of minimal graphs, with respect to subgraph containment, for which Xe (G) > 2.…”

“…It is also known that x|;(G ) < 2 if G lies in one of a few classes of graphs. Chang, Lu, Wu, and Yu [28] showed that xh(G) < 2 for a variety of graph classes, notably if G is bipartite and d-regular for d > 3. Lu, Yu, and Zhang [70] proved that if G is a nice graph which is either 3-connected and bipartite or has minimum degree 5(G) > 8 x(G), then…”

“…The problem of determining conditions under which a graph G has xhiG) < 2 was explored in [28] and [70]. In this chapter, we study the following, more general, problem: Proof First note that, given an edge-weighting which colours vertices by sums, chang ing the sign of every edge weight gives a new edge weighting which also colours vertices by sums.…”