Related papers: Triangle-free geometric intersection graphs with n…

The celebrated Mantel's theorem states that any triangle-free graph on $n$ vertices contains at most $\left\lfloor n^2/4\right\rfloor$ edges. It is natural to ask how many triangles must exist in a graph with more than $\left\lfloor…

A construction of Alon yields a sequence of highly pseudorandom triangle-free graphs with edge density significantly higher than one might expect from comparison with random graphs. We give an alternative construction for such graphs.

We prove that every planar triangle-free graph on $n$ vertices has fractional chromatic number at most $3-\frac{1}{n+1/3}$.

We show that every cubic bridgeless graph with n vertices has at least 3n/4-10 perfect matchings. This is the first bound that differs by more than a constant from the maximal dimension of the perfect matching polytope.

Complete intersections may be unexpectedly simple over fields of positive characteristic: for instance, they may be unirational despite being of general type. One explanation is given by profiles, structure that tracks the special shape of…

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…

Any planar graph has a crossing-free straight-line drawing in the plane. A simultaneous geometric embedding of two n-vertex graphs is a straight-line drawing of both graphs on a common set of n points, such that the edges withing each…

We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma, states that for each $\epsilon>0$ there exists $M$ such that every triangle-free…

The celebrated Hadwiger's conjecture states that if a graph contains no $K_{t+1}$ minor then it is $t$-colourable. If true, it would in particular imply that every $n$-vertex $K_{t+1}$-minor-free graph has an independent set of size at…

Let $k\ge 2$ be an integer and $T_1,\ldots, T_k$ be spanning trees of a graph $G$. If for any pair of vertices $(u,v)$ of $V(G)$, the paths from $u$ to $v$ in each $T_i$, $1\le i\le k$, do not contain common edges and common vertices,…

We prove that a distance-regular graph with intersection array {56,36,9;1,3,48} does not exist. This intersection array is from the table of feasible parameters for distance-regular graphs in "Distance-regular graphs"\ by A.E. Brouwer, A.M.…

Let f(k) denote the maximum such that every simple undirected graph containing two vertices s,t and k edge-disjoint s-t paths, also contains two vertices u,v and f(k) independent u-v paths. Here, a set of paths is independent if none of…

We associate a graph to a possible non-zero zero-divisor in the group algebra of a torsion-free group.

We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight-line edges and crossing-free drawings. This problem has many connections…

We prove that every $n$-vertex planar graph $G$ with no triangle sharing an edge with a 4-cycle has independence ratio $n/\alpha(G) \leq 4 - \varepsilon$ for $\varepsilon = 1/30$. This result implies that the same bound holds for…

A number of famous results about triangle-free graphs determine the maximum chromatic number of graphs of order n with minimum degree greater than n/3. In this note these results are extended to Kr-free graphs.

We show that in the free group of rank 3, given an arbitrary number of automorphisms, the intersection of their fixed subgroups is equal to the fixed subgroup of some other single automorphism.

We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane to so-called \emph{pseudo-simultaneously flippable edges}. Such edges are related to the notion of convex…

There has been much work on the following question: given n how large can a subset of {1,...,n} be that has no arithmetic progressions of length 3. We call such sets 3-free. Most of the work has been asymptotic. In this paper we sketch…

Many real-world networks were found to be highly clustered, and contain a large amount of small cliques. We here investigate the number of cliques of any size k contained in a geometric inhomogeneous random graph: a scale-free network model…