a b s t r a c tA P k -decomposition of a graph G is a set of edge-disjoint paths with k edges that cover the edge set of G. Kotzig (1957) proved that a 3-regular graph admits a P 3 -decomposition if and only if it contains a perfect matching. Kotzig also asked what are the necessary and sufficient conditions for a (2k + 1)-regular graph to admit a decomposition into paths with 2k + 1 edges. We partially answer this question for the case k = 2 by proving that the existence of a perfect matching is sufficient for a triangle-free 5-regular graph to admit a P 5 -decomposition. This result contributes positively to the conjecture of Favaron et al.(2010) that states that every (2k+1)-regular graph with a perfect matching admits a P 2k+1 -decomposition.
Abstract. In 2006, Barát and Thomassen posed the following conjecture: for each tree T , there exists a natural number k T such that, if G is a k T -edge-connected graph and |E(G)| is divisible by |E(T )|, then G admits a decomposition into copies of T . This conjecture was verified for stars, some bistars, paths of length 3, 5, and 2 r for every positive integer r. We prove that this conjecture holds for paths of any fixed length.
A path decomposition of a graph G is a set of edge‐disjoint paths of G that covers the edge set of G. Gallai (1968) conjectured that every connected graph with n vertices admits a path decomposition of size at most ⌊(n+1)∕2⌋. Gallai's conjecture was verified for many classes of graphs. In particular, Lovász (1968) verified this conjecture for graphs with at most one vertex of even degree, and Pyber (1996) verified it for graphs in which every cycle contains a vertex of odd degree. Recently, Bonamy and Perrett verified Gallai's conjecture for graphs with maximum degree at most 5. In this paper, we verify Gallai's conjecture for graphs with treewidth at most 3. Moreover, we show that the only graphs with treewidth at most 3 that do not admit a path decomposition of size at most ⌊n∕2⌋ are isomorphic to K3 or K5−, the graph obtained from K5 by removing an edge.
Tibor Gallai conjectured that the edge set of every connected graph G on n vertices can be partitioned into ⌈n/2⌉ paths. Let G k be the class of all 2k-regular graphs of girth at least 2k − 2 that admit a pair of disjoint perfect matchings. In this work, we show that Gallai's conjecture holds in G k , for every k ≥ 3. Further, we prove that for every graph G in G k on n vertices, there exists a partition of its edge set into n/2 paths of lengths in {2k − 1, 2k, 2k + 1}.
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