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Complement Submodular Information Measures for Balanced and Robust Data Selection A Proof of a Conjecture on Positive and Negative Square Energies of Unicyclic Graphs Laplacian Spectrum of the Weakly Zero-Divisor Graph of a Finite Commutative Ring An identity for second Eulerian numbers via lattice-point counting $t$-tone edge coloring of graphs Constructing Maximal Bumpless Pipedreams for Double Grothendieck Polynomials Mubayi's Polynomial-Ideal Conjecture and Cover-Ideal Turán Methods Implicit Binarization via Complex Phase Dynamics in Combinatorial Optimization The limits of Schur multipliers in Pólya conversion problems for the $q$-permanent function Universality theorems for generalized splines Framing Triangulations for Arbitrary Integer Flow Polytopes On the Common Generalization of Gentle Algebras and Framed Directed Acyclic Graphs The complexity of frugal digraph homomorphisms Chaotic and periodic behavior of jeu de taquin on infinite Young tableaux Enumerating Pattern Avoiding Parking Functions Incidence toric ideals and three-point functions Unique Winning Opening Move in Three-Row Chomp Strong majority colorings of graphs A Balancing Theorem for Spanning Trees of Rectangular Grid Graphs Spectral radius and edge-disjoint connected factors of graphs New invariants for rank metric codes, with applications to the classification of rank two semifields of order 256 Flexible DP-4-coloring of planar graphs without 4-cycles and intersecting triangles Balanced intersection size distributions in projective planes List Reconstruction Problem with List Size Two Is Dimensionality a Barrier for Retrieval Models? The INIEP: Irreducible and Positive Realizations The number of Pfaffian orientations on punctured polygonally cellulated surfaces Explicit Construction of Polytopes whose Ehrhart Polynomials Realize any Given Sign Pattern Finite-state enumeration of adjacency-constrained 132-avoiding permutations AMDS and quantum AMDS Constacyclic codes of length $4p^ς$ over $\mathbb{F}_{{p}^{m}}$
The cycle double cover conjecture from the perspective of...
Jens Walter Fischer · 2023-07-13 · via math.CO updates on arXiv.org

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.