Erdős-Gyárfás conjecture for \(P_8\)-free graphs. (English) Zbl 1498.05142

Summary: A graph is \(P_8\)-free if it contains no induced subgraph isomorphic to the path \(P_8\) on eight vertices. In 1995, Erdős and Gyárfás conjectured that every graph of minimum degree at least three contains a cycle whose length is a power of two. In this paper, we confirm the conjecture for \(P_8\)-free graphs by showing that there exists a cycle of length four or eight in every \(P_8\)-free graph with minimum degree at least three.


05C35 Extremal problems in graph theory
05C38 Paths and cycles
05C07 Vertex degrees


Zbl 0872.05020
Full Text: DOI arXiv


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