Optimal path and cycle decompositions of dense quasirandom graphs

Motivated by longstanding conjectures regarding decompositions of graphs into paths and cycles, we prove the following optimal decomposition results for random graphs. Let $0<p<1$ be constant and let $G\sim G_{n,p}$. Let $odd(G)$ be the number of odd degree vertices in $G$. Then a.a.s. the following hold: (i) $G$ can be decomposed into $\lfloor\Delta(G)/2\rfloor$ cycles and a matching of size $odd(G)/2$. (ii) $G$ can be decomposed into $\max\{odd(G)/2,\lceil\Delta(G)/2\rceil\}$ paths. (iii) $G$ can be decomposed into $\lceil\Delta(G)/2\rceil$ linear forests. Each of these bounds is best possible. We actually derive (i)--(iii) from `quasirandom' versions of our results. In that context, we also determine the edge chromatic number of a given dense quasirandom graph of even order. For all these results, our main tool is a result on Hamilton decompositions of robust expanders by K\"uhn and Osthus.Comment: Some typos from the first version have been correcte

Separation dimension of bounded degree graphs

The 'separation dimension' of a graph $G$ is the smallest natural number $k$ for which the vertices of $G$ can be embedded in $\mathbb{R}^k$ such that any pair of disjoint edges in $G$ can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family $\mathcal{F}$ of total orders of the vertices of $G$ such that for any two disjoint edges of $G$, there exists at least one total order in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. In general, the maximum separation dimension of a graph on $n$ vertices is $\Theta(\log n)$. In this article, we focus on bounded degree graphs and show that the separation dimension of a graph with maximum degree $d$ is at most $2^{9log^{\star} d} d$. We also demonstrate that the above bound is nearly tight by showing that, for every $d$, almost all $d$-regular graphs have separation dimension at least $\lceil d/2\rceil$.Comment: One result proved in this paper is also present in arXiv:1212.675

Linear arboricity of degenerate graphs

A linear forest is a union of vertex-disjoint paths, and the linear arboricity of a graph $G$, denoted by $\operatorname{la}(G)$, is the minimum number of linear forests needed to partition the edge set of $G$. Clearly, $\operatorname{la}(G) \ge \lceil\Delta(G)/2\rceil$ for a graph $G$ with maximum degree $\Delta(G)$. On the other hand, the Linear Arboricity Conjecture due to Akiyama, Exoo, and Harary from 1981 asserts that $\operatorname{la}(G) \leq \lceil(\Delta(G)+1) / 2\rceil$ for every graph $G$. This conjecture has been verified for planar graphs and graphs whose maximum degree is at most $6$, or is equal to $8$ or $10$. Given a positive integer $k$, a graph $G$ is $k$-degenerate if it can be reduced to a trivial graph by successive removal of vertices with degree at most $k$. We prove that for any $k$-degenerate graph $G$, $\operatorname{la}(G) = \lceil\Delta(G)/2 \rceil$ provided $\Delta(G) \ge 2k^2 -k$.Comment: 15 pages, 1 figur

Decomposing cubic graphs into isomorphic linear forests

A common problem in graph colouring seeks to decompose the edge set of a given graph into few similar and simple subgraphs, under certain divisibility conditions. In 1987 Wormald conjectured that the edges of every cubic graph on $4n$ vertices can be partitioned into two isomorphic linear forests. We prove this conjecture for large connected cubic graphs. Our proof uses a wide range of probabilistic tools in conjunction with intricate structural analysis, and introduces a variety of local recolouring techniques.Comment: 49 pages, many figure

Optimal Hamilton covers and linear arboricity for random graphs

In his seminal 1976 paper, P\'osa showed that for all $p\geq C\log n/n$, the binomial random graph $G(n,p)$ is with high probability Hamiltonian. This leads to the following natural questions, which have been extensively studied: How well is it typically possible to cover all edges of $G(n,p)$ with Hamilton cycles? How many cycles are necessary? In this paper we show that for $p\geq C\log n/n$, we can cover $G\sim G(n,p)$ with precisely $\lceil\Delta(G)/2\rceil$ Hamilton cycles. Our result is clearly best possible both in terms of the number of required cycles, and the asymptotics of the edge probability $p$, since it starts working at the weak threshold needed for Hamiltonicity. This resolves a problem of Glebov, Krivelevich and Szab\'o, and improves upon previous work of Hefetz, K\"uhn, Lapinskas and Osthus, and of Ferber, Kronenberg and Long, essentially closing a long line of research on Hamiltonian packing and covering problems in random graphs.Comment: 13 page