Jacques Verstraëte

On Arithmetic Progressions of Cycle Lengths in Graphs

A question recently posed by Haggkvist and Scott asked whether or not there exists a constant c such that, if G is a graph of minimum degree ck, then G contains cycles of k consecutive even lengths. In this paper we answer the question by proving that, for k > 2, a bipartite graph of average degree at least 4k and girth g contains cycles of (g/2 − 1)k consecutive even lengths. We also obtain a short proof of the theorem of Bondy and Simonovits, that a graph of order n and size at least 8(k − 1)n1+1/k has a cycle of length 2k.

A Note on Bipartite Graphs Without 2k-Cycles

The question of the maximum number

Proof Of A Conjecture Of Erdős On Triangles In Set-Systems

\mbox{ex}(m,n,C_{2k})

Turán problems and shadows I

of edges in an m by n bipartite graph without a cycle of length 2k is addressed in this note. For each

On independent sets in hypergraphs

k \geq 2

Rainbow Turán Problems

, it is shown that

Disjoint cycles: integrality gap, hardness, and approximation

\mbox{ex}(m,n,C_{2k}) \leq \begin{cases} (2k-3)\bigl[(mn)^{\frac{k+1}{2k}} + m + n\bigr] & \mbox{ if }k \mbox{ is odd,}\\[2pt] (2k-3)\bigl[m^{\frac{k+2}{2k}}\, n^{\frac{1}{2}} + m + n\bigr] & \mbox{ if }k \mbox{ is even.}\\ \end{cases}

On coupon colorings of graphs

A triangle is a family of three sets A,B,C such that A∩B, B∩C, C∩A are each nonempty, and