Graphs with many r-cliques have large complete r-partite subgraphs

The clique chromatic number of a graph is the minimum number of colours needed to colour its vertices so that no inclusion-wise maximal clique which is not an isolated vertex is monochromatic. We show that every graph of maximum degree $\Delta$ has clique chromatic number $O\left(\frac{\Delta}{\log~\Delta}\right)$. We obtain as a corollary that every $n$-vertex graph has clique chromatic number $O\left(\sqrt{\frac{n}{\log ~n}}\right)$. Both these results are tight.

We give a minimum degree condition sufficent to ensure the existence of a fractional $K_r$-decomposition in a balanced $r$-partite graph (subject to some further simple necessary conditions). This generalises the non-partite problem studied recently by Barber, Lo, K\"uhn, Osthus and the author, and the $3$-partite fractional $K_3$-decomposition problem studied recently by Dukes. Combining our result with recent work by Barber, K\"uhn, Lo, Osthus and Taylor, this gives a minim...

We show that, for $n$ large, there must exist at least \[\frac{n^t}{C^{(1+o(1))t^2}}\] monochromatic $K_t$s in any two-colouring of the edges of $K_n$, where $C \approx 2.18$ is an explicitly defined constant. The old lower bound, due to Erd\H{o}s \cite{E62}, and based upon the standard bounds for Ramsey's theorem, is \[\frac{n^t}{4^{(1+o(1))t^2}}.\]

Let G be a graph on n vertices in which every induced subgraph on s=\log^3 n vertices has an independent set of size at least t=\log n. What is the largest q=q(n) so that every such G must contain an independent set of size at least q ? This is one of several related questions raised by Erdos and Hajnal. We show that q(n)=\Theta(\log^2 n/\log \log n), investigate the more general problem obtained by changing the parameters s and t, and discuss the connection to a related Rams...

Reed and Wood and independently Norine, Seymour, Thomas, and Wollan proved that for each positive integer $t$ there is a constant $c(t)$ such that every graph on $n$ vertices with no $K_t$-minor has at most $c(t)n$ cliques. Wood asked in 2007 if we can take $c(t) = c^t$ for some absolute constant $c$. This question was recently answered affirmatively by Lee and Oum. In this paper, we determine the exponential constant. We prove that every graph on $n$ vertices with no $K_t$-m...

With $\xi_{k}=\xi_{k}^{n,p}$ the number of copies of $K_k$ in the usual (Erd\H{o}s-R\'enyi) random graph $G(n,p)$, $p\geq n^{-2/(k-1)}$ and $\eta>0$, we show when $k>1$ $$\Pr(\xi_k> (1+\eta)\E \xi_k) < \exp [-\gO_{\eta,k} \min\{n^2p^{k-1}\log(1/p), n^kp^{\binom{k}{2}}\}].$$ This is tight up to the value of the constant in the exponent.

Let k_r(n,m) denote the minimum number of r-cliques in graphs with n vertices and m edges. For r=3,4 we give a lower bound on k_r(n,m) that approximates k_r(n,m) with an error smaller than n^r/(n^2-2m). The solution is based on a constraint minimization of certain multilinear forms. In our proof, a combinatorial strategy is coupled with extensive analytical arguments.

All the work made so far on edge-covering a graph by cliques focus on finding the minimum number of cliques that cover the graph. On this paper, we fix the number of cliques that cover a graph by the same number of vertices that the graph has, and give an upper bound for the sum of the number of vertices of these cliques in the cases where this covering is possible.

We study the size of the largest clique $\omega(G(n,\alpha))$ in a random graph $G(n,\alpha)$ on $n$ vertices which has power-law degree distribution with exponent $\alpha$. We show that for `flat' degree sequences with $\alpha>2$ whp the largest clique in $G(n,\alpha)$ is of a constant size, while for the heavy tail distribution, when $0<\alpha<2$, $\omega(G(n,\alpha))$ grows as a power of $n$. Moreover, we show that a natural simple algorithm whp finds in $G(n,\alpha)$ a la...

We show that there is a constant c>0 so that for any fixed r which is at least 3 a.a.s. an r-regular graph on n vertices contains a complete graph on c n^{1/2} vertices as a minor. This confirms a conjecture of Markstrom. Since any minor of an r-regular graph on n vertices has at most rn/2 edges, our bound is clearly best possible up to the value of the constant c. As a corollary, we also obtain the likely order of magnitude of the largest complete minor in a random graph G(n...