435 research outputs found
Color-blind index in graphs of very low degree
Let be an edge-coloring of a graph , not necessarily
proper. For each vertex , let , where is
the number of edges incident to with color . Reorder for
every in in nonincreasing order to obtain , the color-blind
partition of . When induces a proper vertex coloring, that is,
for every edge in , we say that is color-blind
distinguishing. The minimum for which there exists a color-blind
distinguishing edge coloring is the color-blind index of ,
denoted . We demonstrate that determining the
color-blind index is more subtle than previously thought. In particular,
determining if is NP-complete. We also connect
the color-blind index of a regular bipartite graph to 2-colorable regular
hypergraphs and characterize when is finite for a class
of 3-regular graphs.Comment: 10 pages, 3 figures, and a 4 page appendi
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
Vertex-Coloring 2-Edge-Weighting of Graphs
A -{\it edge-weighting} of a graph is an assignment of an integer
weight, , to each edge . An edge weighting naturally
induces a vertex coloring by defining for every
. A -edge-weighting of a graph is \emph{vertex-coloring} if
the induced coloring is proper, i.e., for any edge .
Given a graph and a vertex coloring , does there exist an
edge-weighting such that the induced vertex coloring is ? We investigate
this problem by considering edge-weightings defined on an abelian group.
It was proved that every 3-colorable graph admits a vertex-coloring
-edge-weighting \cite{KLT}. Does every 2-colorable graph (i.e., bipartite
graphs) admit a vertex-coloring 2-edge-weighting? We obtain several simple
sufficient conditions for graphs to be vertex-coloring 2-edge-weighting. In
particular, we show that 3-connected bipartite graphs admit vertex-coloring
2-edge-weighting
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