Chromatic sums of nonseparable simple maps on the plane
The Korean Journal of Computational & Applied Mathematics
Chromatic sums of singular maps on some surfaces
The Korean Journal of Computational & Applied Mathematics
Graph Theory With Applications
Graph Theory With Applications
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Let G(V, E) be a simple graph, and let f be an integer function on V with 1 ≤ f(v) ≤ d(v) to each vertex v ε V. An f-edge cover-coloring of a graph G is a coloring of edge set E such that each color appears at each vertex v ε V at least f(v) times. The f-edge cover chromatic index of G, denoted by χ′fc(G), is the maximum number of colors such that an f-edge cover-coloring of G exists. Any simple graph G has an f-edge cover chromatic index equal to δf or δf - 1, where δf= minv ε V{⌊ d(v)/f(v)⌋} Let G be a connected and not complete graph with χ′fc(G) =δf-1, if for each u, v ε V and e = uv ∉ E, we have χ′fc(G + e) χ′fc(G) then G is called an f-edge covered critical graph. In this paper, some properties on f-edge covered critical graph are discussed. It is proved that if G is an f-edge covered critical graph, then for each u, v ε V and e = uv ∉ E there exists w ε {u,v} with d(w) ≤ δf(f(w)+1) -2 such that w is adjacent to at least d(w) - δf + 1 vertices which are all δf-vertex in G.