A reconfigurations analogue of Brooks’ theorem.

Let G be a simple undirected graph on n vertices with maximum degree Δ. Brooks’ Theorem states that G has a Δ-colouring unless G is a complete graph, or a cycle with an odd number of vertices. To recolour G is to obtain a new proper colouring by changing the colour of one vertex. We show that from a k-colouring, k > Δ, a Δ-colouring of G can be obtained by a sequence of O(n 2) recolourings using only the original k colours unless G is a complete graph or a cycle with an odd number of vertices, or k = Δ + 1, G is Δ-regular and, for each vertex v in G, no two neighbours of v are coloured alike. We use this result to study the reconfiguration graph R k (G) of the k-colourings of G. The vertex set of R k (G) is the set of all possible k-colourings of G and two colourings are adjacent if they differ on exactly one vertex. It is known that if k ≤ Δ(G), then R k (G) might not be connected and it is possible that its connected components have superpolynomial diameter, if k ≥ Δ(G) + 2, then R k (G) is connected and has diameter O(n 2). We complete this structural classification by settling the missing case: if k = Δ(G) + 1, then R k (G) consists of isolated vertices and at most one further component which has diameter O(n 2). We also describe completely the computational complexity classification of the problem of deciding whether two k-colourings of a graph G of maximum degree Δ belong to the same component of R k (G) by settling the case k = Δ(G) + 1. The problem is O(n 2) time solvable for k = 3, PSPACE-complete for 4 ≤ k ≤ Δ(G), O(n) time solvable for k = Δ(G) + 1, O(1) time solvable for k ≥ Δ(G) + 2 (the answer is always yes)

Local colourings and monochromatic partitions in complete bipartite graphs

We show that for any $2$-local colouring of the edges of the balanced complete bipartite graph $K_{n,n}$, its vertices can be covered with at most~$3$ disjoint monochromatic paths. And, we can cover almost all vertices of any complete or balanced complete bipartite $r$-locally coloured graph with $O(r^2)$ disjoint monochromatic cycles.\\ We also determine the $2$-local bipartite Ramsey number of a path almost exactly: Every $2$-local colouring of the edges of $K_{n,n}$ contains a monochromatic path on $n$ vertices.Comment: 18 page

Spanning embeddings of arrangeable graphs with sublinear bandwidth

The Bandwidth Theorem of B\"ottcher, Schacht and Taraz [Mathematische Annalen 343 (1), 175-205] gives minimum degree conditions for the containment of spanning graphs H with small bandwidth and bounded maximum degree. We generalise this result to a-arrangeable graphs H with \Delta(H)<sqrt(n)/log(n), where n is the number of vertices of H. Our result implies that sufficiently large n-vertex graphs G with minimum degree at least (3/4+\gamma)n contain almost all planar graphs on n vertices as subgraphs. Using techniques developed by Allen, Brightwell and Skokan [Combinatorica, to appear] we can also apply our methods to show that almost all planar graphs H have Ramsey number at most 12|H|. We obtain corresponding results for graphs embeddable on different orientable surfaces.Comment: 20 page

Finite reflection groups and graph norms

Given a graph $H$ on vertex set $\{1,2,\cdots, n\}$ and a function $f:[0,1]^2 \rightarrow \mathbb{R}$, define \begin{align*} \|f\|_{H}:=\left\vert\int \prod_{ij\in E(H)}f(x_i,x_j)d\mu^{|V(H)|}\right\vert^{1/|E(H)|}, \end{align*} where $\mu$ is the Lebesgue measure on $[0,1]$. We say that $H$ is norming if $\|\cdot\|_H$ is a semi-norm. A similar notion $\|\cdot\|_{r(H)}$ is defined by $\|f\|_{r(H)}:=\||f|\|_{H}$ and $H$ is said to be weakly norming if $\|\cdot\|_{r(H)}$ is a norm. Classical results show that weakly norming graphs are necessarily bipartite. In the other direction, Hatami showed that even cycles, complete bipartite graphs, and hypercubes are all weakly norming. We demonstrate that any graph whose edges percolate in an appropriate way under the action of a certain natural family of automorphisms is weakly norming. This result includes all previously known examples of weakly norming graphs, but also allows us to identify a much broader class arising from finite reflection groups. We include several applications of our results. In particular, we define and compare a number of generalisations of Gowers' octahedral norms and we prove some new instances of Sidorenko's conjecture.Comment: 29 page

Short rainbow cycles in graphs and matroids

Let $G$ be a simple $n$-vertex graph and $c$ be a colouring of $E(G)$ with $n$ colours, where each colour class has size at least $2$. We prove that $(G,c)$ contains a rainbow cycle of length at most $\lceil \frac{n}{2} \rceil$, which is best possible. Our result settles a special case of a strengthening of the Caccetta-H\"aggkvist conjecture, due to Aharoni. We also show that the matroid generalization of our main result also holds for cographic matroids, but fails for binary matroids.Comment: 9 pages, 2 figure