On the Erdös–Simonovits–Sós Conjecture about the Anti-Ramsey Number of a Cycle

  • Authors:

  • Tao Jiang;Douglas B. West

  • Affiliations:

  • Department of Mathematics and Statistics, Miami University, Oxford, OH 45056, USA (e-mail: jiangt@muohio.edu);Department of Mathematics, University of Illinois, Urbana, IL 61801, USA (e-mail: west@math.uiuc.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2003

Quantified Score

Hi-index 0.00

Visualization

Abstract

Given a positive integer $n$ and a family ${\cal F}$ of graphs, let $f(n,{\cal F})$ denote the maximum number of colours in an edge-colouring of $K_n$ such that no subgraph of $K_n$ belonging to ${\cal F}$ has distinct colours on its edges. Erdös, Simonovits and Sós [6] conjectured for fixed $k$ with $k\geq3$ that $f(n,C_k)\,{=}\, (\frac{k-2}{2}+\frac{1}{k-1})n + O(1)$. This has been proved for $k\leq7$. For general $k$, in this paper we improve the previous bound of $(k-2)n-\big({{k\,{-}\,1}\atop{2}}\big)$ to $f(n,C_k)\leq (\frac{k+1}{2}-\frac{2}{k-1})n - (k-2)$. For even $k$, we further improve it to $\frac{k}{2}n-(k-2)$. We also prove that $f(n,\{C_k,C_{k+1},C_{k+2}\})\leq (\frac{k-2}{2}+\frac{1}{k-1})n-1$, which is sharp.