# Some Results on Palette Index of Cartesian Product Graphs

## DOI:

https://doi.org/10.51408/1963-0070## Keywords:

Edge coloring, Proper edge coloring, Palette, Palette index, Cartesian product, Windmill graph## Abstract

Given a proper edge coloring α of a graph *G*, we define the palette *S _{G}*(

*ν, α*) of a vertex

*ν*∈

*V*(

*G*) as the set of all colors appearing on edges incident to

*ν*. The palette index

*š*(

*G*) of

*G*is the minimum number of distinct palettes occurring in a proper edge coloring of

*G*. The windmill graph

*Wd*(

*n*,

*k*) is an undirected graph constructed for

*k*≥ 2 and

*n*≥ 2 by joining n copies of the complete graph

*K*at a shared universal vertex. In this paper, we determine the bound on the palette index of Cartesian products of complete graphs and simple paths. We also consider the problem of determining the palette index of windmill graphs. In particular, we show that for any positive integers

_{k}*n*,

*k*≥ 2,

*š*(

*Wd*(

*n*, 2

*k*)) =

*n*+ 1.

## References

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## How to Cite

*Mathematical Problems of Computer Science*,

*55*, 26–34. https://doi.org/10.51408/1963-0070

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