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Planar Graphs with Homomorphisms to the 9-cycle

Daniel W. Cranston, Jiaao Li, Zhouningxin Wang, Chunyan Wei

Abstract

We study the problem of finding homomorphisms into odd cycles from planar graphs with high odd-girth. The Jaeger-Zhang conjecture states that every planar graph of odd-girth at least $4k+1$ admits a homomorphism to the odd cycle $C_{2k+1}$. The $k=1$ case is the well-known Grötzsch's $3$-coloring theorem. For general $k$, in 2013 Lovász, Thomassen, Wu, and Zhang showed that it suffices to have odd-girth at least $6k+1$. Improvements are known for $C_5$ and $C_7$ in [Combinatorica 2017, SIDMA 2020, Combinatorica 2022]. For $C_9$ we improve this hypothesis by showing that it suffices to have odd-girth 23. Our main tool is a variation on the potential method applied to modular orientations. This allows more flexibility when seeking reducible configurations. The same techniques also prove some results on circular coloring of signed planar graphs.

Planar Graphs with Homomorphisms to the 9-cycle

Abstract

We study the problem of finding homomorphisms into odd cycles from planar graphs with high odd-girth. The Jaeger-Zhang conjecture states that every planar graph of odd-girth at least admits a homomorphism to the odd cycle . The case is the well-known Grötzsch's -coloring theorem. For general , in 2013 Lovász, Thomassen, Wu, and Zhang showed that it suffices to have odd-girth at least . Improvements are known for and in [Combinatorica 2017, SIDMA 2020, Combinatorica 2022]. For we improve this hypothesis by showing that it suffices to have odd-girth 23. Our main tool is a variation on the potential method applied to modular orientations. This allows more flexibility when seeking reducible configurations. The same techniques also prove some results on circular coloring of signed planar graphs.
Paper Structure (10 sections, 29 theorems, 7 equations, 4 figures)

This paper contains 10 sections, 29 theorems, 7 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Homomorphisms to odd cycles
  3. Modular orientations
  4. Preliminaries
  5. Strongly Z9-connected graphs
  6. Strongly connected (32,B)-orientations of graphs
  7. Proofs of Theorem \ref{['thm:23_homomorphism_to_C9-restated']}$'$ and Theorem \ref{['thm:23_circular_flow_signed_planar_graphs']}
  8. Main theorem
  9. Forbidden configurations
  10. Discharging part

Key Result

Theorem 1.2

Every planar graph of odd-girth at least $23$ admits a homomorphism to $C_{9}$.

Figures (4)

  • Figure 1: The graphs $\alpha K_2$, $T_{a,b,c}$, and multi-$K_4$.
  • Figure 2: Some forbidden configurations in $G$: (a) for \ref{['cla:T117']}, and (b)-(f) for \ref{['cla:configurations_pairs_at most5']}
  • Figure 3: The graphs $T^{o}_{2,2,6}$ in \ref{['cla:T226o']} and $T_{3,3,5}$ in \ref{['cla:T335']}
  • Figure 4: The graphs $Q_{6,6,6,7}^o$ and $Q_{6,6,6,7}^{oo}$ in \ref{['cla:Q6667o']}, and $F$ in \ref{['cla:F1']}

Theorems & Definitions (56)

  • Conjecture 1.1
  • Theorem 1.2
  • Definition 1.3
  • Proposition 1.4
  • Lemma 1.5
  • Definition 1.6
  • proof
  • Definition 1.9
  • Theorem 1.11
  • Theorem 1.12
  • ...and 46 more