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On the $k$-anti-traceability Conjecture

Bin Chen, Stefanie Gerke, Gregory Gutin, Hui Lei, Heis Parker-Cox, Yacong Zhou

Abstract

An oriented graph is called $k$-anti-traceable if the subdigraph induced by every subset with $k$ vertices has a hamiltonian anti-directed path. In this paper, we consider an anti-traceability conjecture. In particular, we confirm this conjecture holds when $k\leq 4$. We also show that every sufficiently large $k$-anti-traceable oriented graph admits an anti-path that contains $n-o(n)$ vertices.

On the $k$-anti-traceability Conjecture

Abstract

An oriented graph is called -anti-traceable if the subdigraph induced by every subset with vertices has a hamiltonian anti-directed path. In this paper, we consider an anti-traceability conjecture. In particular, we confirm this conjecture holds when . We also show that every sufficiently large -anti-traceable oriented graph admits an anti-path that contains vertices.
Paper Structure (5 sections, 12 theorems, 4 equations, 1 figure)

This paper contains 5 sections, 12 theorems, 4 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Values of $f(3)$ and $f(4)$
  3. $f(3)=3$
  4. $f(4)=8$
  5. Almost spanning anti-directed paths

Key Result

Theorem 1.2

(Gr) Every tournament is anti-traceable unless it is isomorphic to one of the tournaments in $\{PT_{3},RT_{5},PT_{7}\}$.

Figures (1)

  • Figure 1: $T_4$

Theorems & Definitions (37)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • Claim 2.2
  • proof
  • Claim 2.3
  • proof
  • proof
  • Theorem 2.5
  • ...and 27 more