Rainbow Induced Subgraphs in Replication Graphs

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.

💡 Research Summary

The paper introduces a Ramsey‑type parameter for graphs that combines the notions of replication and rainbow colourings. Given a graph H, a replication graph G of H is obtained by replacing each vertex of H with a clique (called a replication clique) of arbitrary size and by replacing every edge of H with all possible edges between the two corresponding cliques. In this setting a “rainbow induced subgraph” is an induced copy of H that uses exactly one vertex from each replication clique and whose vertices all receive distinct colours in a proper vertex‑colouring of G. The central quantity studied is

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