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Characterization of graphs without even $F$-orientations

M. Abreu, D. Labbate, F. Romaniello, J. Sheehan

Abstract

A graph $G$ is $1$-extendible if every edge belongs to at least one $1$-factor of $G$. Let $G$ be a graph with a $1$-factor $F$. Then an even $F$-orientation of $G$ is an orientation in which each $F$-alternating cycle has exactly an even number of edges directed in the same fixed direction around the cycle. In this paper, we examine the structure of 1-extendible graphs $G$ which have no even $F$-orientation where $F$ is a fixed $1$-factor of $G$. In the case of graphs of connectivity at least four and k-regular graphs for $k \geq 3$ we give a complete characterization.

Characterization of graphs without even $F$-orientations

Abstract

A graph is -extendible if every edge belongs to at least one -factor of . Let be a graph with a -factor . Then an even -orientation of is an orientation in which each -alternating cycle has exactly an even number of edges directed in the same fixed direction around the cycle. In this paper, we examine the structure of 1-extendible graphs which have no even -orientation where is a fixed -factor of . In the case of graphs of connectivity at least four and k-regular graphs for we give a complete characterization.

Paper Structure

This paper contains 8 sections, 30 theorems, 54 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Structure of 1--extendible graphs
  5. The structure of a minimal counterexample to Theorem \ref{['theorem3.5MAIN']}
  6. Theorem \ref{['theorem5.10']}, Case (i)
  7. Proof of Theorem \ref{['theorem3.5MAIN']}
  8. Proof of Theorem \ref{['Thm2.4']}

Key Result

Lemma 2.3

EOAPG Let $G$ be a graph with a $1$--factor $F$ and an odd zero--sum $F$--set $\cal{C}$$:=\{C_1,\ldots,C_k\}$. Suppose that $C_1,\ldots,C_{k_1}$ are oddly oriented and $C_{k_1+1},\ldots,C_k$ are evenly $F$--oriented in an orientation $\overrightarrow{G}$ of $G$. Let $k_2:=k-k_1$ and $0\le k_i \le k$

Figures (9)

  • Figure 1: The Wagner Graph$W$
  • Figure 2: $\widetilde{G} \notin \cal{W}$ but $H \in \cal{W}$ is an $F$--central subgraph
  • Figure 3: Illustration of Notation \ref{['Not5.1']}
  • Figure 4: Even subdivision of $K_4$
  • Figure 5: Illustration for the proof of Theorem \ref{['Thm2.4']}(ii)
  • ...and 4 more figures

Theorems & Definitions (49)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Corollary 2.4
  • Lemma 2.5
  • Definition 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Definition 3.1
  • Definition 3.2
  • ...and 39 more