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- W2462795702 abstract "Let $G$ be an edge-colored connected graph. A path $P$ in $G$ is called a distance $ell$-proper path if no two edges of the same color appear with fewer than $ell$ edges in between on $P$. The graph $G$ is called $(k,ell)$-proper connected if every pair of distinct vertices of $G$ are connected by $k$ pairwise internally vertex-disjoint distance $ell$-proper paths in $G$. For a $k$-connected graph $G$, the minimum number of colors needed to make $G$ $(k,ell)$-proper connected is called the $(k,ell)$-proper connection number of $G$ and denoted by $pc_{k,ell}(G)$. In this paper, we prove that $pc_{1,2}(G)leq 5$ for any $2$-connected graph $G$. Considering graph operations, we find that $3$ is a sharp upper bound for the $(1,2)$-proper connection number of the join and the Cartesian product of almost all graphs. In addition, we find some basic properties of the $(k,ell)$-proper connection number and determine the values of $pc_{1,ell}(G)$ where $G$ is a traceable graph, a tree, a complete bipartite graph, a complete multipartite graph, a wheel, a cube or a permutation graph of a nontrivial traceable graph." @default.
- W2462795702 created "2016-07-22" @default.
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- W2462795702 date "2016-06-21" @default.
- W2462795702 modified "2023-09-26" @default.
- W2462795702 title "Distance proper connection of graphs" @default.
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- W2462795702 doi "https://doi.org/10.48550/arxiv.1606.06547" @default.
- W2462795702 hasPublicationYear "2016" @default.
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