Vertex Colouring Edge Weightings: a Logarithmic Upper Bound on Weight-Choosability
نویسندگان
چکیده
منابع مشابه
Vertex-Colouring Edge-Weightings
A weighting w of the edges of a graph G induces a colouring of the vertices of G where the colour of vertex v, denoted cv, is ∑ e3v w(e). We show that the edges of every graph that does not contain a component isomorphic to K2 can be weighted from the set {1, . . . , 30} such that in the resulting vertex-colouring of G, for every edge (u, v) of G, cu 6= cv.
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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'...
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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...
متن کاملOn the complexity of vertex-coloring edge-weightings
Given a graph G = (V,E) and a weight function w : E → R, a coloring of vertices of G, induced by w, is defined by χw(v) = ∑ e3v w(e) for all v ∈ V . In this paper, we show that determining whether a particular graph has a weighting of the edges from {1, 2} that induces a proper vertex coloring is NP-complete.
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A weighting of the edges of a graph is called vertexcoloring if the weighted degrees of the vertices yield a proper coloring of the graph. In this paper we show that such a weighting is possible from the weight set {1, 2, 3, 4, 5} for all graphs not containing components with exactly 2 vertices. All graphs in this note are finite and simple. For notation not defined here we refer the reader to ...
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ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2021
ISSN: 1077-8926
DOI: 10.37236/6878