Abstract
A path in an edge-colored graph is called a proper path if no two adjacent edges of the path are colored with one same color. An edge-colored graph is proper connected if any two distinct vertices of the graph are connected by a proper path in the graph. For connected graph G, the smallest number of colors that are needed in order to make G proper connected is called the proper connection number of G, denoted by pc(G). In this paper, we present an upper bound for the proper connection number of a graph in terms of the bridge-block tree of the graph. We also use this upper bound as an efficient tool to investigate the Erdös-Gallai-type problem for the proper connection number of a graph.
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Acknowledgments
The authors are very grateful to the referees for their valuable suggestions and comments which are helpful to improving the paper. Supported by NSFC No.11371205 and 11531011, and PCSIRT.
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Huang, F., Li, X. & Wang, S. Upper bounds of proper connection number of graphs. J Comb Optim 34, 165–173 (2017). https://doi.org/10.1007/s10878-016-0056-2
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DOI: https://doi.org/10.1007/s10878-016-0056-2