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A labeling f : V (G) → {1, 2,. .. , d} of the vertex set of a graph G is said to be proper d-distinguishing if it is a proper coloring of G and any nontrivial automorphism of G maps at least one vertex to a vertex with a different label. The distinguishing chromatic number of G, denoted by χ D (G), is the minimum d such that G has a proper d-distinguishing… (More)

Let G be a connected graph with odd girth 2k + 1. Then G is a (2k + 1)-angulated graph if every two vertices of G are joined by a path such that each edge of the path is in some (2k + 1)-cycle. We prove that if G is (2k + 1)-angulated, and H is connected with odd girth at least 2k + 3, then any retract R of the box (or Cartesian) product G2H is isomorphic… (More)

The concept of a k-pairable graph was introduced by Chen (On k-pairable graphs, Discrete Mathematics 287 (2004), 11–15) as an extension of hypercubes and graphs with an antipodal isomorphism. In this talk, we introduce a special class of k-pairable graphs which are called uniquely pairable graphs. Then we give a characterization of uniquely pairable graphs… (More)

Let denote the class of connected plane bipartite graphs with no pendant edges. A finite face s of a graph G ∈ is said to be a forcing face of G if the subgraph of G obtained by deleting all vertices of s together with their incident edges has exactly one perfect matching. This is a natural generalization of the concept of forcing hexagons in a hexagonal… (More)

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