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Journal ArticleDOI

Perfect path double covers of graphs

01 Apr 1990-Journal of Graph Theory (Wiley)-Vol. 14, Iss: 2, pp 259-272
TL;DR: It is proved that every simple 3-regular graph admits a PPDC consisting of paths of length three, which is equivalent to a perfect path double cover of a graph G on n vertices.
Abstract: A perfect path double cover (PPDC) of a graph G on n vertices is a family of n paths of G such that each edge of G belongs to exactly two members of and each vertex of G occurs exactly twice as an end of a path of . We propose and study the conjecture that every simple graph admits a PPDC. Among other things, we prove that every simple 3-regular graph admits a PPDC consisting of paths of length three.
Citations
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Book ChapterDOI
01 Jan 1990
TL;DR: In this article, the authors introduce a refinement of Seymour's conjecture in which the number of cycles in the cycle double cover plays a significant role, and conjecture that every simple 2-edge-connected graph admits an SCDC.
Abstract: A cycle double cover (CDC) of a graph G is a collection C of cycles of G such that each edge of G belongs to exactly two members of C. P.D. Seymour has conjectured that every 2-edge-connected graph admits a CDC. The purpose of this article is to introduce a refinement of Seymour’s conjecture in which the number of cycles in the CDC plays a significant role. We define a small cycle double cover (SCDC) of a graph G to be a CDC C of G such that ❘C❘ ≤ n - 1, where n is the number of vertices in G, and conjecture that every simple 2-edge-connected graph admits an SCDC. This conjecture and its many ramifications are discussed.

44 citations

Journal IssueDOI
TL;DR: In this article, it was shown that a 171-edge-connected graph has an edge decomposition into paths of length 3 if and only its size is divisible by 3.
Abstract: We prove that a 171-edge-connected graph has an edge-decomposition into paths of length 3 if and only its size is divisible by 3. It is a long-standing problem whether 2-edge-connectedness is sufficient for planar triangle-free graphs, and whether 3-edge-connectedness suffices for graphs in general. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 286–292, 2008

40 citations

Journal ArticleDOI
Hao Li1
TL;DR: It is proved in this paper that every simple graph G admits a perfect path double cover (PPDC), i.e., a set of paths of G such that each edge of G belongs to exactly two of the paths.
Abstract: We prove in this paper that every simple graph G admits a perfect path double cover (PPDC), i.e., a set of paths of G such that each edge of G belongs to exactly two of the paths and each vertex of G is an end of exactly two of the paths, where a path of length zero is considered to have (identical) ends. This was conjectured by A. Bondy in 1988.

29 citations

Journal ArticleDOI
TL;DR: This paper verifies the conjecture that, for any fixed tree T, there exists a natural number kT such that the following holds: If G is a kT-edge-connected graph such that |E(T)| divides |G(G)|, then G has a T-decomposition.
Abstract: Barat and Thomassen have conjectured that, for any fixed tree T, there exists a natural number k T such that the following holds: If G is a k T -edge-connected graph such that |E(T)| divides |E(G)|, then G has a T-decomposition. The conjecture is trivial when T has one or two edges. Before submission of this paper, the conjecture had been verified only for two other trees: the paths of length 3 and 4, respectively. In this paper we verify the conjecture for each path whose length is a power of 2.

28 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that a simple r-regular graph G admits a balanced P4-decomposition if r ≡ 0(mod 3) and G has no cut-edge when r is odd.
Abstract: In this article, we show that every simple r-regular graph G admits a balanced P4-decomposition if r ≡ 0(mod 3) and G has no cut-edge when r is odd. We also show that a connected 4-regular graph G admits a P4-decomposition if and only if |E(G)| ≡ 0(mod 3) by characterizing graphs of maximum degree 4 that admit a triangle-free Eulerian tour. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 135–143, 1999

25 citations

References
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Journal ArticleDOI
TL;DR: Even polyhedral decompositions of a trivalent graph have been shown to have a Tait colouring by three colours a, b, c as mentioned in this paper, which is a strong conjecture.
Abstract: A polyhedral decomposition of a finite trivalent graph G is defined as a set of circuits = {C1, C2, … Cm} with the property that every edge of G occurs exactly twice as an edge of some Ck. The decomposition is called even if every Ck is a simple circuit of even length. If G has a Tait colouring by three colours a, b, c then the (a, b), (b, c) and (c, a) circuits obviously form an even polyhedral decomposition. It is shown that the converse is also true: if G has an even polyhedral decomposition then it also has a Tait colouring. This permits an equivalent formulation of the four colour conjecture (and a much stronger conjecture of Branko Grunbaum) in terms of polyhedral decompositions alone.

201 citations

Journal ArticleDOI
TL;DR: It is shown that for any positive integer s, an s-partition of a graph G = (V, E) is a partition of E into E1 ∪ E2 ∪… ∪ Ek, where∣Ei∣ = s for 1 ≤ i ≤ k − 1 and 1 ≤ ∣Ek∣ ≤ s and each Ei induces a connected subgraph of G.
Abstract: For any positive integer s, an s-partition of a graph G = (V, E) is a partition of E into E1 ∪ E2 ∪… ∪ Ek, where ∣Ei∣ = s for 1 ≤ i ≤ k − 1 and 1 ≤ ∣Ek∣ ≤ s and each Ei induces a connected subgraph of G. We prove (i) If G is connected, then there exists a 2-partition, but not necessarily a 3-partition; (ii) If G is 2-edge connected, then there exists a 3-partition, but not necessarily a 4-partition; (iii) If G is 3-edge connected, then there exists a 4-partition; (iv) If G is 4-edge connected, then there exists an s-partition for all s.

55 citations

Book ChapterDOI
01 Jan 1990
TL;DR: In this article, the authors introduce a refinement of Seymour's conjecture in which the number of cycles in the cycle double cover plays a significant role, and conjecture that every simple 2-edge-connected graph admits an SCDC.
Abstract: A cycle double cover (CDC) of a graph G is a collection C of cycles of G such that each edge of G belongs to exactly two members of C. P.D. Seymour has conjectured that every 2-edge-connected graph admits a CDC. The purpose of this article is to introduce a refinement of Seymour’s conjecture in which the number of cycles in the CDC plays a significant role. We define a small cycle double cover (SCDC) of a graph G to be a CDC C of G such that ❘C❘ ≤ n - 1, where n is the number of vertices in G, and conjecture that every simple 2-edge-connected graph admits an SCDC. This conjecture and its many ramifications are discussed.

44 citations