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Growth rates of the bipartite Erdős-Gyárfás function

Xihe Li, Hajo Broersma, Ligong Wang

Abstract

Given two graphs $G, H$ and a positive integer $q$, an $(H,q)$-coloring of $G$ is an edge-coloring of $G$ such that every copy of $H$ in $G$ receives at least $q$ distinct colors. The bipartite Erdős-Gyárfás function $r(K_{n,n}, K_{s,t}, q)$ is defined to be the minimum number of colors needed for $K_{n,n}$ to have a $(K_{s,t}, q)$-coloring. For balanced complete bipartite graphs $K_{p,p}$, the function $r(K_{n,n}, K_{p,p}, q)$ was studied systematically in [Axenovich, Füredi and Mubayi, {\it J. Combin. Theory Ser. B} {\bf 79} (2000), 66--86]. In this paper, we study the asymptotic behavior of this function for complete bipartite graphs $K_{s,t}$ that are not necessarily balanced. Our main results deal with thresholds and lower and upper bounds for the growth rate of this function, in particular for (sub)linear and (sub)quadratic growth. We also obtain new lower bounds for the balanced bipartite case, and improve several results given by Axenovich, Füredi and Mubayi. Our proof techniques are based on an extension to bipartite graphs of the recently developed Color Energy Method by Pohoata and Sheffer and its refinements, and a generalization of an old result due to Corrádi.

Growth rates of the bipartite Erdős-Gyárfás function

Abstract

Given two graphs and a positive integer , an -coloring of is an edge-coloring of such that every copy of in receives at least distinct colors. The bipartite Erdős-Gyárfás function is defined to be the minimum number of colors needed for to have a -coloring. For balanced complete bipartite graphs , the function was studied systematically in [Axenovich, Füredi and Mubayi, {\it J. Combin. Theory Ser. B} {\bf 79} (2000), 66--86]. In this paper, we study the asymptotic behavior of this function for complete bipartite graphs that are not necessarily balanced. Our main results deal with thresholds and lower and upper bounds for the growth rate of this function, in particular for (sub)linear and (sub)quadratic growth. We also obtain new lower bounds for the balanced bipartite case, and improve several results given by Axenovich, Füredi and Mubayi. Our proof techniques are based on an extension to bipartite graphs of the recently developed Color Energy Method by Pohoata and Sheffer and its refinements, and a generalization of an old result due to Corrádi.

Paper Structure

This paper contains 14 sections, 31 theorems, 24 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Color Energy Method
  4. Proof of Theorem \ref{['th:Kt+']}
  5. Proof of Theorem \ref{['th:C_p']}
  6. An advanced technique
  7. Proofs of Theorems \ref{['th:r=2theta']} and \ref{['th:r3theta']}
  8. Thresholds for linear and quadratic functions
  9. Proofs using and without using the Color Energy Method
  10. Results obtained without using the Color Energy Method
  11. Concluding remarks
  12. Proof of Lemma \ref{['le:UnbalanceQu3+']}
  13. Threshold for $r(K_{n,n}, K_{s,t}, q)=n^2-c$
  14. Threshold for $r(K_{n,n}, K_{s,t}, q)=n^2-O(n)$

Key Result

Theorem 1.1

Let $s, t, q$ be three positive integers.

Figures (1)

  • Figure 1: The graph $\mit\Theta$$(s,t)$ and the $(H\cup \pi(\widetilde{G}))$-reservoir $(\widetilde{R}_A, \widetilde{R}_B, \vec{a}, \vec{b})$.

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 60 more