Lower bound of the hadwiger number of graphs by their average degree

Abstract

The aim of this paper is to show that the minimum Hadwiger number of graphs with average degree k is 0 (k/l/l'~k). Specially, it follows that Hadwiger's conjecture is true for almost all graphs with n vertices, furthermore if k is large enough then for almost all graphs with n vertices and nk edges.

Dense Minors In Graphs Of Large Girth

“…Since an average degree of at least cr √ log r forces a K r minor [ 5,10 ], Theorem 1 has the following consequence: Corollary 2. There exists a constant c ∈ R such that every graph G of girth g(G) 6 log r + 3 loglog r + c and δ(G) 3 has a K r minor.…”

The order of the largest complete minor in a random graph

“…While the influence of the chromatic number on the existence of complete minors is far from clear, the corresponding extremal problem for the average degree has been settled for large complete minors: Thomason [14] asymptotically determined the smallest average degree d(k) which guarantees the existence of a K k minor in any graph of average degree at least d(k). (The order of magnitude k √ log k of d(k) was determined earlier in [8,12].) The extremal graphs are (disjoint copies of) dense random graphs.…”

The minimum number of minimal codewords in an [n,k]-code and in graphic codes

“…Kostochka [13] and Thomason [17] independently proved that a simple graph with p vertices and q edges contains a subgraph which can be contracted into a complete graph with m vertices provided q is at least a constant times pm √ log m. Subsequently, Thomason [18] proved that the condition q > (0.319... + o(1))pm log m suffices (and this is essentially best possible). Proof: We delete successively vertices of degree at most q/p from G until we get a graph G , say, of minimum degree d > p/q = d/2.…”