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The cycle double cover conjecture from the perspective of percolation theory on iterated line graphs

Jens Walter Fischer

TL;DR

The paper tackles the Cycle Double Cover Conjecture for bridgeless triangle-free cubic graphs by lifting edge structures to iterated line graphs and introducing a reduced order two line graph $ rak{L}_2(G)$. A binary labeling on reduced cliques drives the formation of open-edge cycles in $ rak{L}_2(G)$, which project to a double cover of the original graph via a projection $oldsymbol{\pi}$ and an edge-coloring map $oldsymbol{\chi_E}$. Central to the argument is the control of Type A and Type B self-intersections of cycles in the lifted space, achieved through label inversions and a graph of cycles $ rak{G}_{oldsymbol{\Lambda}}$, with a monotone elimination showing Type A intersections can be removed in bridgeless graphs. This yields a cycle decomposition that, upon projection, furnishes a Cycle Double Cover of $G$, advancing a global, percolation-inspired approach to a longstanding combinatorial problem. The work also outlines connections to spin-system viewpoints and poses avenues for oriented covers via half-edge constructions.

Abstract

The cycle double cover conjecture is a long standing problem in graph theory, which links local properties, the valency of a vertex and no bridges, and a global property of the graph, being covered by a particular set of cycles. We prove the conjecture using a lift of walks and cycles in $G$ to sets of open and closed edges on $\mathcal{L}(\mathcal{L}(G))$, the line graph of the line graph of $G$. We exploit that triangles are preserved by the line graph operator to obtain a one-to-one mapping from walks in the underlying graph $G$ to walks on $\mathcal{L}(\mathcal{L}(G))$. We prove that each set of "double walk covers" in $G$ induces a certain set of $\lbrace 0,1\rbrace$ labels on a subgraph covering of $\mathcal{L}(\mathcal{L}(G))$, minus a set of triangles, and conversely, that there is such a set of labels such that its projection back to $G$ implies a double cycle cover, if $G$ is an simple bridgeless triangle-free cubic graph. The techniques applied are inspired by percolation theory, flipping the $\lbrace 0,1\rbrace$ labels to obtain the desired structure.

The cycle double cover conjecture from the perspective of percolation theory on iterated line graphs

TL;DR

The paper tackles the Cycle Double Cover Conjecture for bridgeless triangle-free cubic graphs by lifting edge structures to iterated line graphs and introducing a reduced order two line graph . A binary labeling on reduced cliques drives the formation of open-edge cycles in , which project to a double cover of the original graph via a projection and an edge-coloring map . Central to the argument is the control of Type A and Type B self-intersections of cycles in the lifted space, achieved through label inversions and a graph of cycles , with a monotone elimination showing Type A intersections can be removed in bridgeless graphs. This yields a cycle decomposition that, upon projection, furnishes a Cycle Double Cover of , advancing a global, percolation-inspired approach to a longstanding combinatorial problem. The work also outlines connections to spin-system viewpoints and poses avenues for oriented covers via half-edge constructions.

Abstract

The cycle double cover conjecture is a long standing problem in graph theory, which links local properties, the valency of a vertex and no bridges, and a global property of the graph, being covered by a particular set of cycles. We prove the conjecture using a lift of walks and cycles in to sets of open and closed edges on , the line graph of the line graph of . We exploit that triangles are preserved by the line graph operator to obtain a one-to-one mapping from walks in the underlying graph to walks on . We prove that each set of "double walk covers" in induces a certain set of labels on a subgraph covering of , minus a set of triangles, and conversely, that there is such a set of labels such that its projection back to implies a double cycle cover, if is an simple bridgeless triangle-free cubic graph. The techniques applied are inspired by percolation theory, flipping the labels to obtain the desired structure.
Paper Structure (15 sections, 12 theorems, 10 equations, 36 figures)

This paper contains 15 sections, 12 theorems, 10 equations, 36 figures.

Table of Contents

  1. Introduction & main statements
  2. Structure of reduced order two line graph
  3. Reduced order two line graph and reduced cliques
  4. Type A and Type B self-intersections
  5. Proof of Proposition \ref{['prop:self_ints_everywhere_imply_bridge']}
  6. Reducibility of Type A self-intersections
  7. Generic view of cycles associated to lifted vertex cuts
  8. Type A resolution through $\mathfrak{G}_{\Lambda}$
  9. Monotonicity of number of Type A self-intersections
  10. Choice of connecting cycles through $\mathfrak{G}_{\Lambda}$
  11. Interaction of Type B intersections with generic connnecting cycles
  12. Proof of Theorem \ref{['thm:cycle_double_cover']}
  13. Outlook
  14. Centered vertex-cuts in $\mathcal{L}(G)$ and their lifts
  15. A construction using half edges

Key Result

Theorem 1.2

Let $G=(V,E)$ be a connected simple bridgeless triangle-free cubic graph. Then, there is a cycle double cover such that no edge is covered twice by the same cycle.

Figures (36)

  • Figure 1: Local representation of $\mathfrak{L}_2(G)$ using colors to represent the reduced cliques $\mathbb{X}$. Dashed lines represent edges leading to the remaining parts of the graph. Dotted lines represent the removed triangles.
  • Figure 2: Local representation of $(\mathfrak{L}_2(G),\mathcal{X},\Lambda)$ using colors to represent the closed edges and the open edges inducing cycles. Closed edges are dark red and open edges are the remaining colors. Open edges of identical color belong to the same cycle.
  • Figure 3: The diagram illustrates first the construction of $(\mathfrak{L}_2(G),\mathcal{X},\Lambda)$. Then, subsequently, it shows the projection via $\chi_{\mathcal{E}}$ to a spanning set of closed trails $\mathcal{T}_{\mathcal{L}(G)}$ in $\mathcal{L}(G)$ and, then, via $\pi$ to a set of closed walks $\mathcal{W}_{G}$ in $G$ such that the union over all walks traverses every edge exactly twice.
  • Figure 4: Projection from valid edge-labeling of $\mathcal{L}(G)$ on the left to double coloring of edge of $G$. The arrow indicates the mapping through $\pi$.
  • Figure 5: Two adjacent cycles and a joining reduced clique. Red edges are again closed and cycles are yellow and purple.
  • ...and 31 more figures

Theorems & Definitions (34)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.5
  • Lemma 1.6
  • Definition 1.7
  • Proposition 2.1
  • proof
  • Definition 2.3
  • Lemma 2.4
  • ...and 24 more