MARTIN BACA
Articles written in Proceedings – Mathematical Sciences
Volume 130 All articles Published: 10 January 2020 Article ID 0009 Research Article
Rainbow 2-connectivity of edge-comb product of a cycle and a Hamiltonian graph
MARTIN BACA A N M SALMAN RINOVIA SIMANJUNTAK BETY HAYAT SUSANTI
An edge-colored graph $G$ is rainbow $k$-connected, if for every two verticesof $G$, there are $k$ internally disjoint rainbow paths, i.e., if no two edges of each path are colored the same. The minimum number of colors needed for which there exists a rainbow $k$-connected coloring of $G$, $rc_{k} (G)$, is the rainbow $k$-connection number of $G$. Let $G$ and $H$ be two connected graphs, where $O$ is an orientation of $G$. Let $\vec{e}$ be an oriented edge of $H$. The edge-comb product of $G$ (under the orientation $O$) and $H$ on $\vec{e}$, $G^{o} \vartriangleleft_{\vec{e}} H$, is a graph obtained by taking one copy of $G$ and $|E(G)|$ copies of $H$ and identifying the $i$-th copy of $H$ at the edge $\vec{e}$ to the $i$-th edge of $G$, where the two edges have the same orientation. In this paper, we provide sharp lower and upper bounds for rainbow 2-connection numbers of edge-comb product of a cycle and a Hamiltonian graph. We also determine the rainbow 2-connection numbers of edge-comb product of a cycle with some graphs, i.e. complete graph, fan graph, cycle graph, and wheel graph.
Volume 131, 2021
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