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- W2275126300 abstract "A path in an edge-colored graph is called a proper path if no two adjacent edges of the path receive the same color. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to color its edges, so that every pair of distinct vertices of $G$ is connected by at least one proper path in $G$. Recently, Li and Magnant in [Theory Appl. Graphs 0(1)(2015), Art.2] posed the following conjecture: If $G$ is a connected noncomplete graph of order $n geq 5$ and minimum degree $delta(G) geq n/4$, then $pc(G)=2$. In this paper, we show that this conjecture is true except for two small graphs on 7 and 8 vertices, respectively. As a byproduct we obtain that if $G$ is a connected bipartite graph of order $ngeq 4$ with $delta(G)geq frac{n+6}{8}$, then $pc(G)=2$." @default.
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- W2275126300 date "2016-01-16" @default.
- W2275126300 modified "2023-09-23" @default.
- W2275126300 title "Solution to a conjecture on the proper connection number of graphs" @default.
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- W2275126300 doi "https://doi.org/10.48550/arxiv.1601.04162" @default.
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