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Extremal problems about the order and size of nonhamiltonian locally linear graphs

Feng Liu, Leilei Zhang

Abstract

The study of the relationship between local and global properties of mathematical objects has always been a key subject of investigation in different areas of mathematics. A graph $G$ is called locally linear if the neighbourhood of every vertex of $G$ induces a path. And $G$ is called locally hamiltonian (traceable) if the neighbourhood of every vertex of $G$ induces a hamiltonian (traceable) graph. The local properties of graphs are being studied extensively. For example, the minimum order of a nonhamiltonian (or nontraceable) locally hamiltonian (or traceable) graph has been determined. In this paper, we determine the minimum order of a nonhamiltonian locally linear graph and the minimum size of a nonhamiltonian locally linear graph of a given order.

Extremal problems about the order and size of nonhamiltonian locally linear graphs

Abstract

The study of the relationship between local and global properties of mathematical objects has always been a key subject of investigation in different areas of mathematics. A graph is called locally linear if the neighbourhood of every vertex of induces a path. And is called locally hamiltonian (traceable) if the neighbourhood of every vertex of induces a hamiltonian (traceable) graph. The local properties of graphs are being studied extensively. For example, the minimum order of a nonhamiltonian (or nontraceable) locally hamiltonian (or traceable) graph has been determined. In this paper, we determine the minimum order of a nonhamiltonian locally linear graph and the minimum size of a nonhamiltonian locally linear graph of a given order.
Paper Structure (3 sections, 7 theorems, 67 equations, 4 figures)

This paper contains 3 sections, 7 theorems, 67 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The minimum order of nonhamiltonian locally linear graphs
  3. The minimum size of nonhamiltonian locally traceable graphs

Key Result

Theorem 1

The minimum order of a nonhamiltonian locally linear graph is 12.

Figures (4)

  • Figure 1: The graphs $M_3,M_4,M_5$
  • Figure 2: A nonhamiltonian locally linear graph of order $n$ with maximum vertex degree $n-5$.
  • Figure 3: A connected nonhamiltonian locally linear graph of order $12$.
  • Figure 4: Constructing nonhamiltonian locally linear graphs with $m=2n$.

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4: SMO
  • Lemma 6
  • proof
  • Lemma 9: JDMS
  • Theorem 10: JWTY