Geodesic
Two results on the palette index of graphs
Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 55 (2021) no. 1, pp. 36-43

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Given a proper edge coloring $\alpha$ of a graph $G$, we define the palette $S_G(v,\alpha)$ of a vertex $v\in V(G)$ as the set of all colors appearing on edges incident with $v$. The palette index $\check{s}(G)$ of $G$ is the minimum number of distinct palettes occurring in a proper edge coloring of $G$. A graph $G$ is called nearly bipartite if there exists $ v\in V(G)$ so that $G-v$ is a bipartite graph. In this paper, we give an upper bound on the palette index of a nearly bipartite graph $G$ by using the decomposition of $G$ into cycles. We also provide an upper bound on the palette index of Cartesian products of graphs. In particular, we show that for any graphs $G$ and $H$, $\check{s}(G\square H)\leq \check{s}(G)\check{s}(H)$.
Keywords: edge coloring, proper edge coloring, Cartesian product.
Mots-clés : palette, palette index 
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K. S. Smbatyan. Two results on the palette index of graphs. Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 55 (2021) no. 1, pp. 36-43. http://geodesic.mathdoc.fr/item/UZERU_2021_55_1_a4/