Rainbow Induced Subgraphs in Replication Graphs

TL;DR

This paper investigates the minimal size of replication graphs of a given graph that guarantee rainbow induced subgraphs under any proper coloring, providing bounds, exact values, and experimental insights.

Contribution

It introduces bounds and exact values for the minimal replication graph size ensuring rainbow subgraphs, and presents experimental results and conjectures.

Findings

Bounds for $ ho_R$ for certain graph classes

Exact values of $ ho_R$ for specific graphs

Experimental results and new conjectures

Abstract

A graph $G$ is called a replication graph of a graph $H$ if $G$ is obtained from $H$ by replacing vertices of $H$ by arbitrary cliques of vertices and then replacing each edge in $H$ by all the edges between corresponding cligues. For a given graph $H$ the $\rho_R(H)$ is the minimal number of vertices of a replication graph $G$ of $H$ such that every proper vertex coloring of $G$ contains a rainbow induced subgraph isomorphic to $H$ having exactly one vertex in each replication clique of $G$. We prove some bounds for $\rho_R$ for some classes of graphs and compute some exact values. Also some experimental results obtained by a computer search are presented and conjectures based on them are formulated.