On path decompositions of2k-regular graphs

Abstract

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}.

Gallai's path decomposition conjecture for graphs with treewidth at most 3

“…The proof of the next lemma follows the proof of Lemma 3.2 in . Given a cycle in a graph , a chord of is an edge in that joins two distinct vertices of .…”

Gallai’s path decomposition conjecture for graphs of small maximum degree

“…In 1988, Favaron and Koudier [6] proved that the conjecture holds for graphs where the degree of every vertex is either 2 or 4. More recently, Botler and Jiménez [3] proved that the conjecture holds for 2k-regular graphs of large girth and admitting a pair of disjoint perfect matchings. Jiménez and Wakabayashi [7] showed that the conjecture holds for a subclass of planar, triangle-free graphs satisfying a distance condition on the vertices of odd degree.…”

Path and cycle decompositions of dense graphs

“…, for i ∈ [k] and j ∈ [7]. For each i ∈ [k] and j ∈ [7], we construct a set F P CA i,j of fictive edges as above and let F P CA be the union of these sets.…”