Rainbow induced subgraphs in proper vertex colorings

TL;DR

This paper investigates the minimum size of graphs that guarantee rainbow induced subgraphs of a given graph in any proper coloring, providing bounds, exact values for certain classes, and exploring related combinatorial problems.

Contribution

It introduces bounds and exact values for the rainbow subgraph parameter (H), advancing understanding of rainbow structures in proper colorings and their computational aspects.

Findings

Established bounds for (H)

Computed exact values for specific graph classes

Explored combinatorial problems related to paths

Abstract

For a given graph $H$ we define $\rho(H)$ to be the minimum order of a graph $G$ such that every proper vertex coloring of $G$ contains a rainbow induced subgraph isomorphic to $H$. We give upper and lower bounds for $\rho(H)$, compute the exact value for some classes of graphs, and consider an interesting combinatorial problem connected with computation of $\rho(H)$ for paths. This research is motivated by some ideas in on-line graph coloring algorithms.