Red-blue clique partitions and (1-1)-transversals

Motivated by the problem of Gallai on $(1-1)$-transversals of $2$-intervals, it was proved by the authors in 1969 that if the edges of a complete graph $K$ are colored with red and blue (both colors can appear on an edge) so that there is no monochromatic induced $C_4$ and $C_5$ then the vertices of $K$ can be partitioned into a red and a blue clique. Aharoni, Berger, Chudnovsky and Ziani recently strengthened this by showing that it is enough to assume that there is no induced monochromatic $C_4$ and there is no induced $C_5$ in {\em one of the colors}. Here this is strengthened further, it is enough to assume that there is no monochromatic induced $C_4$ and there is no $K_5$ on which both color classes induce a $C_5$. We also answer a question of Kaiser and Rabinovich, giving an example of six $2$-convex sets in the plane such that any three intersect but there is no $(1-1)$-transversal for them

On-Line Competitive Coloring Algorithms

In the area of on-line algorithms the existence of competitive algorithms is one of the most frequently studied questions. However, competitive on-line graph coloring algorithms exist only for very restricted families of graphs. We introduce a new concept, called on-line competitive algorithms. Our main problem is whether on-line competitive coloring algorithms exist for all classes of graphs. This concept can be useful if somebody must design an on-line coloring algorithm and the input graph is only known to be in a specified class of graphs. In this case the designer want to get the best algorithm but this is usually hard. An on-line competitive algorithm offers less: it comes together with a function f such that for every graph in the class the number of colors it uses can be bounded by f(Ø (G)) where Ø (G) is the minimum number of colors can be achieved at all for that graph by any on-line algorithm (that algorithm may know the graph in advance). Of course, the smaller is f ..

Random hypergraph irregularity

A hypergraph is k-irregular if there is no set of k vertices all of which have the same degree. We asymptotically determine the probability that a random uniform hypergraph is k-irregular

Graphs with maximal number of minimum cuts

SIGLEAvailable at INIST (FR), Document Supply Service, under shelf-number : 17660, issue : a.1992 n.880-M / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc