Non-three-colorable common graphs exist

A graph H is called common if the total number of copies of H in every graph and its complement asymptotically minimizes for random graphs. A former conjecture of Burr and Rosta, extending a conjecture of Erdos asserted that every graph is common. Thomason disproved both conjectures by showing that the complete graph of order four is not common. It is now known that in fact the common graphs are very rare. Answering a question of Sidorenko and of Jagger, Stovicek and Thomason from 1996 we show that the 5-wheel is common. This provides the first example of a common graph that is not three-colorable.Comment: 9 page

Branching rules, Kostka-Foulkes polynomials and qq-multiplicities in tensor product for the root systems B_n,C_nB\_{n},C\_{n} and D_nD\_{n}

The Kostka-Foulkes polynomials $K$ related to a root system $\phi$ can be defined as alternated sums running over the Weyl group associated to $\phi .$ By restricting these sums over the elements of the symmetric group when $$ is of type $B,C$ or $D$, we obtain again a class $\widetilde{K}$ of Kostka-Foulkes polynomials. When $\phi$ is of type $C$ or $D$ there exists a duality beetween these polynomials and some natural $q$-multiplicities $U$ in tensor product \cite{lec}. In this paper we first establish identities for the $\widetilde{K}$ which implies in particular that they can be decomposed as sums of Kostka-Foulkes polynomials related to the root system of type $A$ with nonnegative integer coefficients. Moreover these coefficients are branching rule coefficients. This allows us to clarify the connection beetween the $q$-multiplicities $U$ and the polynomials defined by Shimozono and Zabrocki in \cite{SZ}. Finally we establish that the $q$-multiplicities $U$ defined for the tensor powers of the vector representation coincide up to a power of $q$ with the one dimension sum $X$ introduced in \cite{Ok} This shows that in this case the one dimension sums $$ are affine Kazhdan-Lusztig polynomials

An approximate version of Sidorenko's conjecture

A beautiful conjecture of Erd\H{o}s-Simonovits and Sidorenko states that if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. This conjecture also has an equivalent analytic form and has connections to a broad range of topics, such as matrix theory, Markov chains, graph limits, and quasirandomness. Here we prove the conjecture if H has a vertex complete to the other part, and deduce an approximate version of the conjecture for all H. Furthermore, for a large class of bipartite graphs, we prove a stronger stability result which answers a question of Chung, Graham, and Wilson on quasirandomness for these graphs.Comment: 12 page

Graphs with few 3-cliques and 3-anticliques are 3-universal

For given integers k, l we ask whether every large graph with a sufficiently small number of k-cliques and k-anticliques must contain an induced copy of every l-vertex graph. Here we prove this claim for k=l=3 with a sharp bound. A similar phenomenon is established as well for tournaments with k=l=4.Comment: 12 pages, 1 figur

The step Sidorenko property and non-norming edge-transitive graphs

Sidorenko's Conjecture asserts that every bipartite graph H has the Sidorenko property, i.e., a quasirandom graph minimizes the density of H among all graphs with the same edge density. We study a stronger property, which requires that a quasirandom multipartite graph minimizes the density of H among all graphs with the same edge densities between its parts; this property is called the step Sidorenko property. We show that many bipartite graphs fail to have the step Sidorenko property and use our results to show the existence of a bipartite edge-transitive graph that is not weakly norming; this answers a question of Hatami [Israel J. Math. 175 (2010), 125-150].Comment: Minor correction on page