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Characterizations of Graphs Having Large Proper Connection Numbers

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Lumduanhom C., Laforge E., Zhang P.

Discussiones Mathematicae Graph Theory

2016

tom 36

nr 2

439-453

Let G be an edge-colored connected graph. A path P is a proper path in G if no two adjacent edges of P are colored the same. If P is a proper u − v path of length d(u, v), then P is a proper u − v geodesic. An edge coloring c is a proper-path coloring of a connected graph G if every pair u, v of distinct vertices of G are connected by a proper u − v path in G, and c is a strong proper-path coloring if every two vertices u and v are connected by a proper u− v geodesic in G. The minimum number of colors required for a proper-path coloring or strong proper-path coloring of G is called the proper connection number pc(G) or strong proper connection number spc(G) of G, respectively. If G is a nontrivial connected graph of size m, then pc(G) ≤ spc(G) ≤ m and pc(G) = m or spc(G) = m if and only if G is the star of size m. In this paper, we determine all connected graphs G of size m for which pc(G) or spc(G) is m − 1,m − 2 or m − 3.

Ograniczanie wyników

1 Discussiones Mathematicae Graph Theory

1 Laforge E.

1 Lumduanhom C.

1 Zhang P.

1 2016

Wyniki wyszukiwania

Characterizations of Graphs Having Large Proper Connection Numbers