Universal graphs with a forbidden subgraph: Block path solidity

TL;DR

This paper characterizes the structure of finite connected graphs C that admit a countable universal C-free graph with a block path structure, showing that such graphs must have complete blocks, extending previous results.

Contribution

It generalizes a known result by Furedi and Komjath to a broader class of graphs with block path structures, establishing conditions for the existence of universal C-free graphs.

Findings

Blocks of C are complete graphs.

Generalizes previous results on universal C-free graphs.

Fits into broader conjectures on countable C-free graphs.

Abstract

Let C be a finite connected graph for which there is a countable universal C-free graph, and whose tree of blocks is a path. Then the blocks of C are complete. This generalizes a result of Furedi and Komjath, and fits naturally into a set of conjectures regarding the existence of countable C-free graphs, with C an arbitrary finite connected graph.