Gallai's path decomposition conjecture for cartesian product of graphs (\uppercase\expandafter{\romannumeral 2})

Let $G$ be a graph of order $n$. A path decomposition $\mathcal{P}$ of $G$ is a collection of edge-disjoint paths that covers all the edges of $G$. Let $p(G)$ denote the minimum number of paths needed in a path decomposition of $G$. Gallai conjectured that if $G$ is connected, then $p(G)\leq \lceil\frac{n}{2}\rceil$. In this paper, we prove that Gallai's path decomposition conjecture holds for the cartesian product $G\Box H$, where $H$ is any graph and $G$ is a unicyclic graph or a bicyclic graph.Comment: 26 pages, 2 figures. arXiv admin note: text overlap with arXiv:2310.1118

Path and cycle decompositions of dense graphs

We make progress on three long standing conjectures from the 1960s about path and cycle decompositions of graphs. Gallai conjectured that any connected graph on $n$ vertices can be decomposed into at most $\left\lceil \frac{n}{2}\right\rceil$ paths, while a conjecture of Haj\'{o}s states that any Eulerian graph on $n$ vertices can be decomposed into at most $\left\lfloor \frac{n-1}{2}\right\rfloor$ cycles. The Erd\H{o}s-Gallai conjecture states that any graph on $n$ vertices can be decomposed into $O(n)$ cycles and edges. We show that if $G$ is a sufficiently large graph on $n$ vertices with linear minimum degree, then the following hold. (i) $G$ can be decomposed into at most $\frac{n}{2}+o(n)$ paths. (ii) If $G$ is Eulerian, then it can be decomposed into at most $\frac{n}{2}+o(n)$ cycles. (iii) $G$ can be decomposed into at most $\frac{3 n}{2}+o(n)$ cycles and edges. If in addition $G$ satisfies a weak expansion property, we asymptotically determine the required number of paths/cycles for each such $G$. (iv) $G$ can be decomposed into $\max \left\{\frac{odd(G)}{2},\frac{\Delta(G)}{2}\right\}+o(n)$ paths, where $odd(G)$ is the number of odd-degree vertices of $G$. (v) If $G$ is Eulerian, then it can be decomposed into $\frac{\Delta(G)}{2}+o(n)$ cycles. All bounds in (i)-(v) are asymptotically best possible.Comment: 48 pages, 2 figures; final version, to appear in the Journal of the London Mathematical Societ

On path-cycle decompositions of triangle-free graphs

In this work, we study conditions for the existence of length-constrainedpath-cycle decompositions, that is, partitions of the edge set of a graph intopaths and cycles of a given minimum length. Our main contribution is thecharacterization of the class of all triangle-free graphs with odd distance atleast $3$ that admit a path-cycle decomposition with elements of length atleast $4$. As a consequence, it follows that Gallai's conjecture on pathdecomposition holds in a broad class of sparse graphs

On path-cycle decompositions of triangle-free graphs

In this work, we study conditions for the existence of length-constrained path-cycle decompositions, that is, partitions of the edge set of a graph into paths and cycles of a given minimum length. Our main contribution is the characterization of the class of all triangle-free graphs with odd distance at least $3$ that admit a path-cycle decomposition with elements of length at least $4$. As a consequence, it follows that Gallai's conjecture on path decomposition holds in a broad class of sparse graphs