We introduce a new approach and prove that the maximum number of triangles in C 5 -free graph on n vertices is at most ( 1 + o ) 3 2 ? . show connection to r-uniform hypergraphs without (Berge) cycles length less than six, estimate their possible size. Using our approach, we also (slightly) improve previous size an induced- 4 graph.

How dense can every induced subgraph of b nc vertices (0 < 1) of a triangle-free graph of order n be? Tools will be developed to estimate the local density of graphs, based on the spectrum of the graph and on a fractional viewpoint. These tools are used to refute a conjecture of Erd} os et.al. about the local density of triangle-free graphs for a certain range of , by estimating the local densi...

An H-free editing problem asks whether we can edit at most k edges to make a graph contain no induced copy of the fixed graph H. We obtain a polynomial kernel for this problem when H is a diamond. The incompressibility dichotomy for H being a 3-connected graph [4] and the classical complexity dichotomy [1] suggest that except for H being a complete/empty graph, Hfree editing problems admit poly...

In this paper, we show that if G is an l-connected claw-free graph with minimum degree at least three and l ∈ {2, 3}, then for any maximum independent set S, there exists a 2-factor in which each cycle contains at least l − 1 vertices in S.

a subset $s$ of vertices in a graph $g$ is called a geodetic set if every vertex not in $s$ lies on a shortest path between two vertices from $s$‎. ‎a subset $d$ of vertices in $g$ is called dominating set if every vertex not in $d$ has at least one neighbor in $d$‎. ‎a geodetic dominating set $s$ is both a geodetic and a dominating set‎. ‎the geodetic (domination‎, ‎geodetic domination) number...

A complete graph is the graph in which every two vertices are adjacent. For a graph G = (V,E), the complete width of G is the minimum k such that there exist k independent sets Ni ⊆ V , 1 ≤ i ≤ k, such that the graph G obtained from G by adding some new edges between certain vertices inside the sets Ni, 1 ≤ i ≤ k, is a complete graph. The complete width problem is to decide whether the complete...

Given a fixed graph H, the H-free editing problem asks whether we can edit at most k edges to make contain no induced copy of H. We obtain polynomial kernel for this when H is diamond. The incompressibility dichotomy being 3-connected and classical complexity suggest that except complete/empty graph, problems admit kernels only few small graphs Therefore, believe our result an essential step to...