Fábio Botler

Decompositions of triangle-free 5-regular graphs into paths of length five

A 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 ( 2 k + 1 ) -regular graph to admit a decomposition into paths with 2 k + 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? ( 2 k + 1 ) -regular graph with a perfect matching admits a P 2 k + 1 -decomposition.

Decomposing highly connected graphs into paths of length five

Abstract Barat and Thomassen (2006) posed the following decomposition 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 . In a series of papers, Thomassen verified this conjecture for stars, some bistars, paths of length 3, and paths whose length is a power of 2. We verify this conjecture for paths of length 5.

Path decompositions of regular graphs with prescribed girth

A Pl-decomposition of a graph G is a set of pairwise edge-disjoint paths of G with l edges that cover the edge set of G. Kotzig [Kotzig, A., Aus der Theorie der endlichen regularen Graphen dritten und vierten Grades, Casopis Pěst. Mat. 82 (1957), pp. 76–92.] proved that a 3-regular graph admits a P3-decomposition if and only if it contains a perfect matching, and also asked what are the necessary and sufficient conditions for an l-regular graph to admit a Pl-decomposition, for odd l. Let g, l and m be positive integers with g≥3. We prove that, (i) if l is odd and m>2⌊(l−2)/(g−2)⌋, then every ml-regular graph with girth at least g that contains an m-factor admits a Pl-decomposition; (ii) if m>⌊(l−2)/(g−2)⌋, then every 2ml-regular graph with girth at least g admits a Pl-decomposition.

Decompositions of highly connected graphs into paths of any given length

Abstract We study the decomposition conjecture posed by Barat and Thomassen (2006), which states that, for each tree T, there exists a natural number k T such that, if G is a k T -edge-connected graph and | E ( T ) | divides | E ( G ) | , then G admits a partition of its edge set into classes each of which induces a copy of T. In a series of papers, starting in 2008, Thomassen has verified this conjecture for stars, some bistars, paths of length 3, and paths whose length is a power of 2. In 2014, we verified this conjecture for paths of length 5. In this paper we verify this conjecture for paths of any given length.

A constrained path decomposition of cubic graphs and the path number of cacti

Kotzig (1957) proved that a cubic graph has a perfect matching if and only if it has a 3-path decomposition (that is, a partition of the edge set into paths of length 3). This result was generalized by Jaeger, Payan, and Kouider (1983), who proved that a (2k + l)-regular graph with a perfect matching can be decomposed into bistars. (A bistar is a graph obtained from two disjoint stars by joining their centers with an edge.) In another direction, Heinrich, Liu and Yu (1999) proved that a 3m-regular graph G admits a balanced 3-path decomposition if and only if G contains an m-factor.

Decomposing 8-regular graphs into paths of length 4

A

On path decompositions of 2k-regular graphs☆

T

Gallai's conjecture for graphs with treewidth 3

-decomposition of a graph

Decomposing highly edge-connected graphs into paths of any given length ☆

G

Decompositions of highly connected graphs into paths of length five

is a set of edge-disjoint copies of