Generalized Kneser coloring theorems with combinatorial proofs

The Kneser conjecture (1955) was proved by Lov\'asz (1978) using the Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its extensions. Only in 2000, Matou\v{s}ek provided the first combinatorial proof of the Kneser conjecture. Here we provide a hypergraph coloring theorem, with a combinatorial proof, which has as special cases the Kneser conjecture as well as its extensions and generalization by (hyper)graph coloring theorems of Dol'nikov, Alon-Frankl-Lov\'asz, Sarkaria, and Kriz. We also give a combinatorial proof of Schrijver's theorem.Comment: 19 pages, 4 figure

A Combinatorial Proof of Kneser'sConjecture*

Kneser's conjecture, first proved by Lovász in 1978, states that the graph with all k-element subsets of {1, 2, . . . , n} as vertices and with edges connecting disjoint sets has chromatic number n−2k+2. We derive this result from Tucker's combinatorial lemma on labeling the vertices of special triangulations of the octahedral ball. By specializing a proof of Tucker's lemma, we obtain self-contained purely combinatorial proof of Kneser's conjectur

Investigations in the semi-strong product of graphs and bootstrap percolation

The semi-strong product of graphs G and H is a way of forming a new graph from the graphs G and H. The vertex set of the semi-strong product is the Cartesian product of the vertex sets of G and H, V(G) x V(H). The edges of the semi-strong product are determined as follows: (g1,h1)(g2,h2) is an edge of the product whenever g1g2 is an edge of G and h1h2 is an edge of H or g1 = g2 and h1h2 is an edge of H. A natural subject for investigation is to determine properties of the semi-strong product in terms of those properties of its factors. We investigate distance, independence, matching, and domination in the semi-strong product Bootstrap Percolation is a process defined on a graph. We begin with an initial set of infected vertices. In each subsequent round, uninfected vertices become infected if they are adjacent to at least r infected vertices. Once infected, vertices remain infected. The parameter r is called the percolation threshold. When G is finite, the infection either stops at a proper subset of G or all of V(G) becomes infected. If all of V(G) eventually becomes infected, then we say that the infection percolates and we call the initial set of infected vertices a percolating set. The cardinality of a minimum percolating set of G with percolation threshold r is denoted m(G,r). We determine m(G,r) for certain Kneser graphs and bipartite Kneser graphs