Related papers: Fractional triangle decompositions in almost compl…

Let the Andr\'{a}sfai graph $\mathrm{And}_k$ be defined as the graph with vertex set $\{v_0,v_1,...c, v_{3k-2}\}$ and two vertices $v_i$ and $v_j$ being adjacent iff $|i-j| \equiv 1\mod 3$. The graphs $\mathrm{And}_k$ are maximal…

A conjecture due to the fourth author states that every $d$-regular planar multigraph can be $d$-edge-coloured, provided that for every odd set $X$ of vertices, there are at least $d$ edges between $X$ and its complement. For $d = 3$ this…

A fullerene graph is a cubic bridgeless plane graph with all faces of size 5 and 6. We show that that every fullerene graph on n vertices can be made bipartite by deleting at most sqrt{12n/5} edges, and has an independent set with at least…

A well-known theorem of Mantel states that every $n$-vertex graph with more than $\lfloor n^2/4\rfloor $ edges contains a triangle. An interesting problem in extremal graph theory studies the minimum number of edges contained in triangles…

Balogh, Bar\'at, Gerbner, Gy\'arf\'as, and S\'ark\"ozy proposed the following conjecture. Let $G$ be a graph on $n$ vertices with minimum degree at least $3n/4$. Then for every $2$-edge-colouring of $G$, the vertex set $V(G)$ may be…

We study triangle decompositions of graphs. We consider constructions of classes of graphs where every edge lies on a triangle and the addition of the minimum number of multiple edges between already adjacent vertices results in a strongly…

We compute a minimum degree threshold sufficient for 3-partite graphs to admit a fractional triangle decomposition. Together with recent work of Barber, K\"uhn, Lo, Osthus and Taylor, this leads to bounds for exact decompositions and in…

A typical decomposition question asks whether the edges of some graph $G$ can be partitioned into disjoint copies of another graph $H$. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the…

We obtain some $d\ge2$ such that every graph $G$ with no induced copy of the five-vertex path $P_5$ has at most $\alpha(G)\omega(G)^d$ vertices. This ``off-diagonal Ramsey'' statement implies that every such graph $G$ has fractional…

A new, constructive proof with a small explicit constant is given to the Erd\H{o}s-Pyber theorem which says that the edges of a graph on $n$ vertices can be partitioned into complete bipartite subgraphs so that every vertex is covered at…

We prove that for any integer $k\geq 2$ and $\varepsilon>0$, there is an integer $\ell_0\geq 1$ such that any $k$-uniform hypergraph on $n$ vertices with minimum codegree at least $(1/2+\varepsilon)n$ has a fractional decomposition into…

Turan's Theorem states that every graph of a certain edge density contains a complete graph $K^k$ and describes the unique extremal graphs. We give a similar Theorem for l-partite graphs. For large l, we find the minimal edge density…

Our main result is that every graph $G$ on $n\ge 10^4r^3$ vertices with minimum degree $\delta(G) \ge (1 - 1 / 10^4 r^{3/2} ) n$ has a fractional $K_r$-decomposition. Combining this result with recent work of Barber, K\"uhn, Lo and Osthus…

We study the typical structure and the number of triangle-free graphs with $n$ vertices and $m$ edges where $m$ is large enough so that a typical triangle-free graph has a cut containing nearly all of its edges, but may not be bipartite.…

We prove a decomposition theorem for the class of triangle-free graphs that do not contain a subdivision of the complete graph on four vertices as an induced subgraph. We prove that every graph of girth at least~5 in this class is…

Paul Erd\H{o}s suggested the following problem: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. Here we show that the number of maximal triangle-free graphs is at most $2^{n^2/8+o(n^2)}$, which matches the…

A graph G is (a:b)-colorable if there exists an assignment of b-element subsets of {1,...,a} to vertices of G such that sets assigned to adjacent vertices are disjoint. We first show that for every triangle-free planar graph G and a vertex…

We show that for all $\ell, k, n$ with $\ell \leq k/2$ and $(k-\ell)$ dividing $n$ the following hypergraph-variant of Lehel's conjecture is true. Every $2$-edge-colouring of the $k$-uniform complete hypergraph $\mathcal{K}_n^{(k)}$ on $n$…

Let $G = (V,E)$ be a simple graph and let $\{R,B\}$ be a partition of $E$. We prove that whenever $|E| + \min\{ |R|, |B| \} > { |V| \choose 2 }$, there exists a subgraph of $G$ isomorphic to $K_3$ which contains edges from both $R$ and $B$.…

An embedding of a graph in $3$-space is linkless if for every two disjoint cycles there exists an embedded ball that contains one of the cycles and is disjoint from the other. We prove that every bipartite linklessly embeddable (simple)…