Choosability and paintability of the lexicographic product of graphs

TL;DR

This paper investigates the choice and paint numbers of the lexicographic product of graphs, providing bounds based on maximum degree and properties of component graphs, advancing understanding of graph coloring complexities.

Contribution

It establishes new upper bounds for the choice and paint numbers of lexicographic graph products in terms of maximum degree and component graph parameters.

Findings

Bound on choice number: ch(G[H]) ≤ (4Δ+2)(ch(H) + log₂ n)

Bound on paint number: χ_P(G[H]) ≤ (4Δ+2)(χ_P(H) + log₂ n)

Results apply to graphs with maximum degree Δ and n vertices in H.

Abstract

This paper studies the choice number and paint number of the lexicographic product of graphs. We prove that if $G$ has maximum degree $\Delta$, then for any graph $H$ on $n$ vertices $ch(G[H]) \le (4\Delta+2)(ch(H) +\log_2 n)$ and $\chi_P(G[H]) \le (4\Delta+2) (\chi_P(H)+ \log_2 n)$.