The sum of degrees in cliques

We investigate lower bounds on the average degree in r-cliques in graphs of order n and size greater than t(r,n), where t(r,n) is the size of the Turan graph on n vertices and r color classes. Continuing earlier research of Edwards and Faudree, we completely prove a conjecture of Bollobas and Erdoes from 1975.Comment: 10 page

Maximal Chordal Subgraphs

A chordal graph is a graph with no induced cycles of length at least $4$. Let $f(n,m)$ be the maximal integer such that every graph with $n$ vertices and $m$ edges has a chordal subgraph with at least $f(n,m)$ edges. In 1985 Erd\H{o}s and Laskar posed the problem of estimating $f(n,m)$. In the late '80s, Erd\H{o}s, Gy\'arf\'as, Ordman and Zalcstein determined the value of $f(n,n^2/4+1)$ and made a conjecture on the value of $f(n,n^2/3+1)$. In this paper we prove this conjecture and answer the question of Erd\H{o}s and Laskar, determining $f(n,m)$ asymptotically for all $m$ and exactly for $m \leq n^2/3+1$