On the path partition of graphs

Let $G$ be a graph of order $n$. The maximum and minimum degree of $G$ are denoted by $\Delta$ and $\delta$ respectively. The \emph{path partition number} $\mu (G)$ of a graph $G$ is the minimum number of paths needed to partition the vertices of $G$. Magnant, Wang and Yuan conjectured that $\mu (G)\leq \max \left \{ \frac{n}{\delta +1}, \frac{\left( \Delta -\delta \right) n}{\left( \Delta +\delta \right) }\right \} .$ In this work, we give a positive answer to this conjecture, for $\Delta \geq 2 \delta$.\medskip \end{abstract}Comment: 13 pages,3 figure

Stability for vertex cycle covers

In 1996 Kouider and Lonc proved the following natural generalization of Dirac’s Theorem: for any integer k > 2, if G is an n-vertex graph with minimum degree at least n/k, then there are k − 1 cycles in G that together cover all the vertices. This is tight in the sense that there are n-vertex graphs that have minimum degree n/k − 1 and that do not contain k − 1 cycles with this property. A concrete example is given by In,k = Kn \ K(k−1)n/k+1 (an edge-maximal graph on n vertices with an independent set of size (k − 1)n/k + 1). This graph has minimum degree n/k − 1 and cannot be covered with fewer than k cycles. More generally, given positive integers k1, . . . , kr summing to k, the disjoint union Ik1n/k,k1 +· · ·+Ikrn/k,kr is an n-vertex graph with the same properties. In this paper, we show that there are no extremal examples that differ substantially from the ones given by this construction. More precisely, we obtain the following stability result: if a graph G has n vertices and minimum degree nearly n/k, then it either contains k − 1 cycles covering all vertices, or else it must be close (in ‘edit distance’) to a subgraph of Ik1n/k,k1 + · · · + Ikrn/k,kr , for some sequence k1, . . . , kr of positive integers that sum to k. Our proof uses Szemer´edi’s Regularity Lemma and the related machinery

Cycle partitions of regular graphs

Magnant and Martin conjectured that the vertex set of any $d$-regular graph $G$ on $n$ vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$, improving a result of Han, who showed that in this range almost all vertices of $G$ can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact, our proof gives a partition of $V(G)$ into cycles. We also show that, if $d = \Omega(n)$ and $G$ is bipartite, then $V(G)$ can be partitioned into $n / (2d)$ paths (this bound in tight for bipartite graphs).Comment: 31 pages, 1 figur