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On some path-critical Ramsey numbers

Ye Wang, Yanyan Song

Abstract

For graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $r$ such that any red-blue edge coloring of $K_r$ contains a red $G$ or a blue $H$. The path-critical Ramsey number $R_π(G,H)$ is the largest $n$ such that any red-blue edge coloring of $K_r \setminus P_{n}$ contains a red $G$ or a blue $H$, where $r=R(G,H)$ and $P_{n}$ is a path of order $n$. In this note, we show a general upper bound for $R_π(G,H)$, and determine the exact values for some cases of $R_π(G,H)$.

On some path-critical Ramsey numbers

Abstract

For graphs and , the Ramsey number is the smallest such that any red-blue edge coloring of contains a red or a blue . The path-critical Ramsey number is the largest such that any red-blue edge coloring of contains a red or a blue , where and is a path of order . In this note, we show a general upper bound for , and determine the exact values for some cases of .
Paper Structure (2 sections, 11 theorems, 10 equations)

This paper contains 2 sections, 11 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Proofs

Key Result

Theorem 1

For connected graph $G$ of order $n$ and graph $H$ with $\Delta(G)=n-1$ and $n \geq s(H)$, if $r=R(G, H) \leq(\chi(H)-1)n+s(H)-1$, then In particular, if $G$ is $H$-good, then $R_{\pi}(G,H)\leq n-1.$

Theorems & Definitions (15)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Claim 1
  • ...and 5 more