Spectral radius conditions for fractional [a,b][a,b]-covered graphs

A graph $G$ is called fractional $[a,b]$-covered if for every edge $e$ of $G$ there is a fractional $[a,b]$-factor with the indicator function $h$ such that $h(e)=1$. In this paper, we provide tight spectral radius conditions for graphs being fractional $[a,b]$-covered.Comment: 9 page

A bandwidth theorem for approximate decompositions

We provide a degree condition on a regular $n$-vertex graph $G$ which ensures the existence of a near optimal packing of any family $\mathcal H$ of bounded degree $n$-vertex $k$-chromatic separable graphs into $G$. In general, this degree condition is best possible. Here a graph is separable if it has a sublinear separator whose removal results in a set of components of sublinear size. Equivalently, the separability condition can be replaced by that of having small bandwidth. Thus our result can be viewed as a version of the bandwidth theorem of B\"ottcher, Schacht and Taraz in the setting of approximate decompositions. More precisely, let $\delta_k$ be the infimum over all $\delta\ge 1/2$ ensuring an approximate $K_k$-decomposition of any sufficiently large regular $n$-vertex graph $G$ of degree at least $\delta n$. Now suppose that $G$ is an $n$-vertex graph which is close to $r$-regular for some $r \ge (\delta_k+o(1))n$ and suppose that $H_1,\dots,H_t$ is a sequence of bounded degree $n$-vertex $k$-chromatic separable graphs with $\sum_i e(H_i) \le (1-o(1))e(G)$. We show that there is an edge-disjoint packing of $H_1,\dots,H_t$ into $G$. If the $H_i$ are bipartite, then $r\geq (1/2+o(1))n$ is sufficient. In particular, this yields an approximate version of the tree packing conjecture in the setting of regular host graphs $G$ of high degree. Similarly, our result implies approximate versions of the Oberwolfach problem, the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree.Comment: Final version, to appear in the Proceedings of the London Mathematical Societ

Fractional clique decompositions of dense graphs

For each $r\ge 4$, we show that any graph $G$ with minimum degree at least $(1-1/100r)|G|$ has a fractional $K_r$-decomposition. This improves the best previous bounds on the minimum degree required to guarantee a fractional $K_r$-decomposition given by Dukes (for small $r$) and Barber, K\"uhn, Lo, Montgomery and Osthus (for large $r$), giving the first bound that is tight up to the constant multiple of $r$ (seen, for example, by considering Tur\'an graphs). In combination with work by Glock, K\"uhn, Lo, Montgomery and Osthus, this shows that, for any graph $F$ with chromatic number $\chi(F)\ge 4$, and any $\varepsilon>0$, any sufficiently large graph $G$ with minimum degree at least $(1-1/100\chi(F)+\varepsilon)|G|$ has, subject to some further simple necessary divisibility conditions, an (exact) $F$-decomposition.Comment: 15 pages, 1 figure, submitte

On sufficient conditions for Hamiltonicity in dense graphs

We study structural conditions in dense graphs that guarantee the existence of vertex-spanning substructures such as Hamilton cycles. It is easy to see that every Hamiltonian graph is connected, has a perfect fractional matching and, excluding the bipartite case, contains an odd cycle. Our main result in turn states that any large enough graph that robustly satisfies these properties must already be Hamiltonian. Moreover, the same holds for embedding powers of cycles and graphs of sublinear bandwidth subject to natural generalisations of connectivity, matchings and odd cycles. This solves the embedding problem that underlies multiple lines of research on sufficient conditions for Hamiltonicity in dense graphs. As applications, we recover and establish Bandwidth Theorems in a variety of settings including Ore-type degree conditions, P\'osa-type degree conditions, deficiency-type conditions, locally dense and inseparable graphs, multipartite graphs as well as robust expanders

Edge-decompositions of graphs with high minimum degree

A fundamental theorem of Wilson states that, for every graph $F$, every sufficiently large $F$-divisible clique has an $F$-decomposition. Here a graph $G$ is $F$-divisible if $e(F)$ divides $e(G)$ and the greatest common divisor of the degrees of $F$ divides the greatest common divisor of the degrees of $G$, and $G$ has an $F$-decomposition if the edges of $G$ can be covered by edge-disjoint copies of $F$. We extend this result to graphs $G$ which are allowed to be far from complete. In particular, together with a result of Dross, our results imply that every sufficiently large $K_3$-divisible graph of minimum degree at least $9n/10+o(n)$ has a $K_3$-decomposition. This significantly improves previous results towards the long-standing conjecture of Nash-Williams that every sufficiently large $K_3$-divisible graph with minimum degree at least $3n/4$ has a $K_3$-decomposition. We also obtain the asymptotically correct minimum degree thresholds of $2n/3 +o(n)$ for the existence of a $C_4$-decomposition, and of $n/2+o(n)$ for the existence of a $C_{2\ell}$-decomposition, where $\ell\ge 3$. Our main contribution is a general `iterative absorption' method which turns an approximate or fractional decomposition into an exact one. In particular, our results imply that in order to prove an asymptotic version of Nash-Williams' conjecture, it suffices to show that every $K_3$-divisible graph with minimum degree at least $3n/4+o(n)$ has an approximate $K_3$-decomposition,Comment: 41 pages. This version includes some minor corrections, updates and improvement

Fractional total colourings of graphs of high girth

Reed conjectured that for every epsilon>0 and Delta there exists g such that the fractional total chromatic number of a graph with maximum degree Delta and girth at least g is at most Delta+1+epsilon. We prove the conjecture for Delta=3 and for even Delta>=4 in the following stronger form: For each of these values of Delta, there exists g such that the fractional total chromatic number of any graph with maximum degree Delta and girth at least g is equal to Delta+1

Structure of Cubic Lehman Matrices

A pair $(A,B)$ of square $(0,1)$-matrices is called a \emph{Lehman pair} if $AB^T=J+kI$ for some integer $k\in\{-1,1,2,3,\ldots\}$. In this case $A$ and $B$ are called \emph{Lehman matrices}. This terminology arises because Lehman showed that the rows with the fewest ones in any non-degenerate minimally nonideal (mni) matrix $M$ form a square Lehman submatrix of $M$. Lehman matrices with $k=-1$ are essentially equivalent to \emph{partitionable graphs} (also known as $(\alpha,\omega)$-graphs), so have been heavily studied as part of attempts to directly classify minimal imperfect graphs. In this paper, we view a Lehman matrix as the bipartite adjacency matrix of a regular bipartite graph, focusing in particular on the case where the graph is cubic. From this perspective, we identify two constructions that generate cubic Lehman graphs from smaller Lehman graphs. The most prolific of these constructions involves repeatedly replacing suitable pairs of edges with a particular $6$-vertex subgraph that we call a $3$-rung ladder segment. Two decades ago, L\"{u}tolf \& Margot initiated a computational study of mni matrices and constructed a catalogue containing (among other things) a listing of all cubic Lehman matrices with $k =1$ of order up to $17 \times 17$. We verify their catalogue (which has just one omission), and extend the computational results to $20 \times 20$ matrices. Of the $908$ cubic Lehman matrices (with $k=1$) of order up to $20 \times 20$, only two do not arise from our $3$-rung ladder construction. However these exceptions can be derived from our second construction, and so our two constructions cover all known cubic Lehman matrices with $k=1$