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Erdős-Gyárfás conjecture on graphs without long induced paths

Anand Shripad Hegde, R. B. Sandeep, P. Shashank

Abstract

Erdős and Gyárfás conjectured in 1994 that every graph with minimum degree at least 3 has a cycle of length a power of 2. In 2022, Gao and Shan (Graphs and Combinatorics) proved that the conjecture is true for $P_8$-free graphs, i.e., graphs without any induced copies of a path on 8 vertices. In 2024, Hu and Shen (Discrete Mathematics) improved this result by proving that the conjecture is true for $P_{10}$ -free graphs. With the aid of a computer search, we improve this further by proving that the conjecture is true for $P_{13}$ -free graphs.

Erdős-Gyárfás conjecture on graphs without long induced paths

Abstract

Erdős and Gyárfás conjectured in 1994 that every graph with minimum degree at least 3 has a cycle of length a power of 2. In 2022, Gao and Shan (Graphs and Combinatorics) proved that the conjecture is true for -free graphs, i.e., graphs without any induced copies of a path on 8 vertices. In 2024, Hu and Shen (Discrete Mathematics) improved this result by proving that the conjecture is true for -free graphs. With the aid of a computer search, we improve this further by proving that the conjecture is true for -free graphs.

Paper Structure

This paper contains 1 section, 4 theorems, 2 figures, 1 table.

Key Result

lemma thmcounterlemma

Let $G^*$ be a minimal counterexample to Erdős-Gyárfás conjecture, such that $G^*$ is $P_k$-free for an integer $k\geq 3$. Let $\{v_0,v_1,\ldots,v_{n^*-1}\}$ be the set of vertices of $G^*$. Let $G$ be a graph with the vertex set $\{v_0,v_1,\ldots,v_{n-1}\}$, where $3\leq n\leq n^*$, such that the f Then explore($G,k$) returns False.

Figures (2)

  • Figure 1: The algorithm
  • Figure 2: Markström graph: the unique smallest cubic planar graph having no 4-cycle and no 8-cycle, but having a 16-cycle. The bold (purple) edges show a 16-cycle.

Theorems & Definitions (6)

  • lemma thmcounterlemma
  • proof
  • corollary thmcountercorollary
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem