Some undecidable problems involving the edge-coloring and vertex-coloring 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 3-edge-weighting of some graphs

Vertex-coloring 2-edge-weighting of graphs

A k-edge-weighting w of a graph G is an assignment of an integer weight, w(e) ∈ {1, . . . , k}, to each edge e. An edge weighting naturally induces a vertex coloring c by defining c(u) = ∑ u∼e w(e) for every u ∈ V (G). A k-edge-weighting of a graph G is vertexcoloring if the induced coloring c is proper, i.e., c(u) ≠ c(v) for any edge uv ∈ E(G). Given a graph G and a vertex coloring c0, does th...

Adjacent vertex-distinguishing edge coloring of graphs

An adjacent vertex-distinguishing edge coloring, or avd-coloring, of a graph G is a proper edge coloring of G such that no pair of adjacent vertices meets the same set of colors. Let mad(G) and ∆(G) denote the maximum average degree and the maximum degree of a graph G, respectively. In this paper, we prove that every graph G with ∆(G) ≥ 5 and mad(G) < 3− 2 ∆ can be avd-colored with ∆(G) + 1 col...

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...