Vertex-coloring 2-edge-weighting of graphs

Edge-coloring Vertex-weightings of Graphs

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $n$. A $k$-vertex weightings of a graph $G$ is a mapping $w: V(G) to {1, ldots, k}$. A $k$-vertex weighting induces an edge labeling $f_w: E(G) to N$ such that $f_w(uv)=w(u)+w(v)$. Such a labeling is called an {it edge-coloring k-vertex weightings} if $f_{w}(e)not= f_{w}(echr(chr(chr('39')39chr('39'))39chr(chr('39')39chr('39'...

Vertex-Coloring Edge-Weighting of Bipartite Graphs with Two Edge Weights

It was proved that every 3-connected bipartite graph admits a vertex-coloring S-edge-weighting for S = {1, 2} (H. Lu, Q. Yu and C. Zhang, Vertex-coloring 2-edge-weighting of graphs, European J. Combin., 32 (2011), 22-27). In this paper, we show that every 2-connected and 3-edge-connected bipartite graph admits a vertex-coloring S-edgeweighting for S ∈ {{0, 1}, {1, 2}}. These bounds we obtain ar...

Vertex-coloring edge-weightings of graphs

A k-edge-weighting of a graph G is a mapping w : E(G) → {1, 2, . . . , k}. An edgeweighting w induces a vertex coloring fw : V (G) → N defined by fw(v) = ∑ v∈e w(e). An edge-weighting w is vertex-coloring if fw(u) 6= fw(v) for any edge uv. The current paper studies the parameter μ(G), which is the minimum k for which G has a vertexcoloring k-edge-weighting. Exact values of μ(G) are determined f...

A practical algorithm for [r, s, t]-coloring of graph

Coloring graphs is one of important and frequently used topics in diverse sciences. In the majority of the articles, it is intended to find a proper bound for vertex coloring, edge coloring or total coloring in the graph. Although it is important to find a proper algorithm for graph coloring, it is hard and time-consuming too. In this paper, a new algorithm for vertex coloring, edge coloring an...

Vertex-coloring 3-edge-weighting of some graphs