Abstract A long-standing conjecture of Erd?s and Simonovits asserts that for every rational number $r\in (1,2)$ there exists a bipartite graph H such $\mathrm{ex}(n,H)=\Theta(n^r)$ . So far this is known to be true only rationals form $1+1/k$ $2-1/k$ , integers $k\geq 2$ In paper, we add new which the true: $2-2/(2k+1)$ This in turn also gives an affirmative answer question Pinchasi Sharir on c...

We prove the Černý conjecture for one-cluster automata with prime length cycle. Consequences are given for the hybrid Roadcoloring-Černý conjecture for digraphs with a proper cycle of prime length.

In this paper we study prime graphs of finite groups. The graph a group $G$, also known as the Gruenberg-Kegel graph, is with vertex set {primes dividing $|G|$} and an edge $p$-$q$ if only there exists element order $pq$ in $G$. theory, studying has been important topic for past almost half century. Only recently solvable groups have characterized theoretical terms only. paper, continue line re...

let $g $ be a finite group and let $gamma(g)$ be the prime graph‎ ‎of g‎. ‎assume $2 < q = p^{alpha} < 100$‎. ‎we determine finite groups‎ ‎g such that $gamma(g) = gamma(u_3(q))$ and prove that if $q neq‎ ‎3‎, ‎5‎, ‎9‎, ‎17$‎, ‎then $u_3(q)$ is quasirecognizable by prime graph‎, ‎i.e‎. ‎if $g$ is a finite group with the same prime graph as the‎ ‎finite simple group $u_3(q)$‎, ‎then $g$ has a un...

let l := u3(11). in this article, we classify groups with the same order and degree pattern as an almost simple group related to l. in fact, we prove that l, l:2 and l:3 are od-characterizable, and l:s3 is 5-fold od-characterizable.

A graph G is called normal if there exist two coverings, C and S of its vertex set such that every member induces a clique in G, an independent C∩S≠∅ for C∈C S∈S. It has been conjectured by De Simone Körner 1999 does not contain C5, C7 C7‾ as induced subgraph. We disprove this conjecture.