570 research outputs found
Distinguishing colorings of graphs and their subgraphs
In this paper, several distinguishing colorings of graphs are studied, such as vertex distinguishing proper edge coloring, adjacent vertex distinguishing proper edge coloring, vertex distinguishing proper total coloring, adjacent vertex distinguishing proper total coloring. Finally, some related chromatic numbers are determined, especially the comparison of the correlation chromatic numbers between the original graph and the subgraphs are obtained
Graph Colorings with Constraints
A graph is a collection of vertices and edges, often represented by points and connecting lines in the plane. A proper coloring of the graph assigns colors to the vertices, edges, or both so that proximal elements are assigned distinct colors. Here we examine results from three different coloring problems. First, adjacent vertex distinguishing total colorings are proper total colorings such that the set of colors appearing at each vertex is distinct for every pair of adjacent vertices. Next, vertex coloring total weightings are an assignment of weights to the vertices and edges of a graph so that every pair of adjacent vertices have distinct weight sums. Finally, edge list multi-colorings consider assignments of color lists and demands to edges; edges are colored with a subset of their color list of size equal to its color demand so that adjacent edges have disjoint sets. Here, color sets consisting of measurable sets are considered
Asymmetric -colorings of graphs
We show that the edges of every 3-connected planar graph except can be
colored with two colors in such a way that the graph has no color preserving
automorphisms. Also, we characterize all graphs which have the property that
their edges can be -colored so that no matter how the graph is embedded in
any orientable surface, there is no homeomorphism of the surface which induces
a non-trivial color preserving automorphism of the graph
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
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