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is a Bound on the Adjacent Vertex Distinguishing Edge Chromatic Number
An adjacent vertex distinguishing edge-coloring or an \avd-coloring of a
simple graph is a proper edge-coloring of such that no pair of adjacent
vertices meets the same set of colors. We prove that every graph with maximum
degree and with no isolated edges has an \avd-coloring with at most
colors, provided that
Group twin coloring of graphs
For a given graph , the least integer such that for every
Abelian group of order there exists a proper edge labeling
so that for each edge is called the \textit{group twin
chromatic index} of and denoted by . This graph invariant is
related to a few well-known problems in the field of neighbor distinguishing
graph colorings. We conjecture that for all graphs
without isolated edges, where is the maximum degree of , and
provide an infinite family of connected graph (trees) for which the equality
holds. We prove that this conjecture is valid for all trees, and then apply
this result as the base case for proving a general upper bound for all graphs
without isolated edges: , where
denotes the coloring number of . This improves the best known
upper bound known previously only for the case of cyclic groups
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