We present Filter-Kruskal – a simple modification of Kruskal’s algorithm that avoids sorting edges that are “obviously” not in the MST. For arbitrary graphs with random edge weights, Filter-Kruskal runs in time O ( m + n log n log m n ) , i.e. in linear time for not too sparse graphs. Experiments indicate that the algorithm has very good practical performance over the entire range of edge densi...

A bound on consecutive clique numbers of graphs is established. This bound is evaluated and shown to often be much better than the bound of the Kruskal-Katona theorem. A bound on non-consecutive clique numbers is also proven.

As we have seen, antichains and intersecting families are fundamental to Extremal Set Theory. The two central theorems, Sperner’s Theorem and the Erdős–Ko–Rado Theorem, have inspired decades of research since their discovery, helping establish Extremal Set Theory as a vibrant and rapidly growing area of Discrete Mathematics. One must, then, pay a greater than usual amount of respect to the Krus...

for some N ≥ k + 1, then the equality in (1.2) occurs only when a = 0. Kruskal-Macaulay functions are relevant for their applications to the study of antichains in multisets (see for example [11, 2]), posets, rings and polyhedral combinatorics (see [5] and the survey [3]). In particular, they play and important role in proving results, extensions and generalizations of classical problems concer...