9,953 research outputs found
An Introduction to Fuzzy Edge Coloring
In this paper, a new concept of fuzzy edge coloring is introduced. The fuzzy edge coloring is an assignment of colors to edges of a fuzzy graph G. It is proper if no two strong adjacent edges of G will receive the same color. Fuzzy edge chromatic number of G is least positive integer for which G has a proper fuzzy edge coloring. In this paper, the fuzzy edge chromatic number of different classes of fuzzy graphs and the fuzzy edge chromatic number of fuzzy line graphs are found. Isochromatic fuzzy graph is also defined
Adjacent vertex-distinguishing proper edge-coloring of strong product of graphs
Let G be a finite, simple, undirected and connected graph. The adjacent vertex-distinguishing proper edge-coloring is the minimum number of colors required for a proper edge-coloring of G, in which no two adjacent vertices are incident to edges colored with the same set of colors. The minimum number of colors required for an adjacent vertex-distinguishing proper edgecoloring of G is called the adjacent vertex-distinguishing proper edge-chromatic index. In this paper, I compute adjacent vertex-distinguishing proper edge-chromatic index of strong product of graphs
Normal edge-colorings of cubic graphs
A normal -edge-coloring of a cubic graph is an edge-coloring with
colors having the additional property that when looking at the set of colors
assigned to any edge and the four edges adjacent it, we have either exactly
five distinct colors or exactly three distinct colors. We denote by
the smallest , for which admits a normal
-edge-coloring. Normal -edge-colorings were introduced by Jaeger in order
to study his well-known Petersen Coloring Conjecture. More precisely, it is
known that proving for every bridgeless cubic graph is
equivalent to proving Petersen Coloring Conjecture and then, among others,
Cycle Double Cover Conjecture and Berge-Fulkerson Conjecture. Considering the
larger class of all simple cubic graphs (not necessarily bridgeless), some
interesting questions naturally arise. For instance, there exist simple cubic
graphs, not bridgeless, with . On the other hand, the known
best general upper bound for was . Here, we improve it by
proving that for any simple cubic graph , which is best
possible. We obtain this result by proving the existence of specific no-where
zero -flows in -edge-connected graphs.Comment: 17 pages, 6 figure
A Strong Edge-Coloring of Graphs with Maximum Degree 4 Using 22 Colors
In 1985, Erd\H{o}s and Ne\'{s}etril conjectured that the strong edge-coloring
number of a graph is bounded above by when is even and
when is odd. They gave a simple
construction which requires this many colors. The conjecture has been verified
for . For , the conjectured bound is 20. Previously,
the best known upper bound was 23 due to Horak. In this paper we give an
algorithm that uses at most 22 colors.Comment: 9 pages, 4 figure
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