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• # Artykuł - szczegóły

## Discussiones Mathematicae Graph Theory

2016 | 36 | 2 | 439-453

## Characterizations of Graphs Having Large Proper Connection Numbers

EN

### Abstrakty

EN
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.

EN

439-453

wydano
2016-05-01
otrzymano
2014-10-15
poprawiono
2015-08-05
zaakceptowano
2015-08-10
online
2016-04-15

### Twórcy

autor
• Department of Mathematics Srinakharinwirot University Sukhumvit Soi 23, Bangkok, 10110, Thailand
autor
• Department of Mathematics Western Michigan University Kalamazoo, MI 49008, USA
autor
• Department of Mathematics Western Michigan University Kalamazoo, MI 49008, USA