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- W2616264412 abstract "A path in an edge-colored graph is called conflict-free if it contains a color that is used by exactly one of its edges. An edge-colored graph G is conflict-free connected if for any two distinct vertices of G, there is a conflict-free path connecting them. For a connected graph G, the conflict-free connection number of G, denoted by cfc(G), is defined as the minimum number of colors that are required to make G conflict-free connected. In this paper, we investigate the conflict-free connection numbers of connected claw-free graphs, especially line graphs. We use L(G) to denote the line graph of a graph G. In general, the k-iterated line graph of a graph G, denoted by $$L^k(G)$$ , is the line graph of the graph $$L^{k-1}(G)$$ , where $$kge 2$$ is a positive integer. We first show that for an arbitrary connected graph G, there exists a positive integer k such that $$cfc(L^k(G))le 2$$ . Secondly, we get the exact value of the conflict-free connection number of a connected claw-free graph, especially a connected line graph. Thirdly, we prove that for an arbitrary connected graph G and an arbitrary positive integer k, we always have $$cfc(L^{k+1}(G))le cfc(L^k(G))$$ , with only the exception that G is isomorphic to a star of order at least 5 and $$k=1$$ . Finally, we obtain the exact values of $$cfc(L^k(G))$$ , and use them as an efficient tool to get the smallest nonnegative integer $$k_0$$ such that $$cfc(L^{k_0}(G))=2$$ ." @default.
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- W2616264412 date "2017-01-01" @default.
- W2616264412 modified "2023-09-27" @default.
- W2616264412 title "Conflict-Free Connection Numbers of Line Graphs" @default.
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- W2616264412 doi "https://doi.org/10.1007/978-3-319-71150-8_14" @default.
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