相关论文: A proof of the Cycle Double Cover Conjecture
In this paper, a proof of the cycle double cover conjecture is presented. The cycle double cover conjecture purports that if a graph is bridgeless, then there exists a list of cycles in the graph such that every edge in the graph appears in…
Given a bridgeless graph G, the well-known cycle double cover conjecture posits that there is a list of cycles of G, such that every edge appears in exactly two cycles. In this paper, we prove the cycle double cover conjecture. More…
The cycle double cover conjecture states that a graph is bridge-free if and only if there is a family of edge-simple cycles such that each edge is contained in exactly two of them. It was formulated independently by Szekeres (1973) and…
The strong cycle double cover conjecture states that for every circuit $C$ of a bridgeless cubic graph $G$, there is a cycle double cover of $G$ which contains $C$. We conjecture that there is even a 5-cycle double cover $S$ of $G$ which…
The Cycle double cover (CDC) conjecture states that for every bridgeless graph $G$, there exists a family $\mathcal{F}$ of cycles such that each edge of the graph is contained in exactly two members of $\mathcal{F}$. Given an embedding of a…
In this paper, for each graph G, a free edge set F is defined. To study the existence of cycle double cover, the naive cycle double cover of G and F have been defined and studied. In the main theorem, the paper, based on the Kuratowski…
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…
We explore the well-known Jaeger's directed cycle double cover conjecture which is equivalent to the assertion that every cubic bridgeless graph has an embedding on a closed orientable surface with no dual loop. We associate each cubic…
A directed cycle double cover of a graph G is a family of cycles of G, each provided with an orientation, such that every edge of G is covered by exactly two oppositely directed cycles. Explicit obstacles to the existence of a directed…
Which $2$-regular subgraph $R$ of a cubic graph $G$ can be extended to a cycle double cover of $G$? We provide a condition which ensures that every $R$ satisfying this condition is part of a cycle double cover of $G$. As one consequence, we…
We show that every $2$-connected cubic graph $G$ has a cycle double cover if $G$ has a spanning subgraph $F$ such that (i) every component of $F$ has an even number of vertices (ii) every component of $F$ is either a cycle or a subdivision…
Given two graphs, a mapping between their edge-sets is cycle-continuous, if the preimage of every cycle is a cycle. The motivation for this notion is Jaeger's conjecture that for every bridgeless graph there is a cycle-continuous mapping to…
A circuit double cover of a bridgeless graph is a collection of even subgraphs such that every edge is contained in exactly two subgraphs of the given collection. Such a circuit double cover describes an embedding of the corresponding graph…
In this article, we give a positive answer to the cycle double cover conjecture. Ones who are mainly interesting in the proof of the conjecture can only read Sections 2 and 4.
A conjecture of Berge suggests that every bridgeless cubic graph can have its edges covered with at most five perfect matchings. Since three perfect matchings suffice only when the graph in question is $3$-edge-colourable, the rest of cubic…
The Berge-Fulkerson conjecture states that every bridgeless cubic graph can be covered with six perfect matchings such that each edge is covered exactly twice. An equivalent reformulation is that it's possible to find a 6-cycle 4-cover. In…
Bermond, Jackson and Jaeger [{\em J. Combin. Theory Ser. B} 35 (1983): 297-308] proved that every bridgeless ordinary graph $G$ has a circuit $4$-cover and Fan [{\em J. Combin. Theory Ser. B} 54 (1992): 113-122] showed that $G$ has a…
Berge Conjecture states that every bridgeless cubic graph has 5 perfect matchings such that each edge is contained in at least one of them. In this paper, we show that Berge Conjecture holds for two classes of cubic graphs, cubic graphs…
We show that every bridgeless cubic graph $G$ with $m$ edges has a cycle cover of length at most $1.6 m$. Moreover, if $G$ does not contain any intersecting circuits of length $5$, then $G$ has a cycle cover of length $212/135 \cdot m…
We study a counting version of Cycle Double Cover Conjecture. We discuss why it is more interesting to count circuits (i.e., graphs isomorphic to $C_k$ for some $k$) instead of cycles (graphs with all degrees even). We give an…