On the minimum of the Hadwiger number for graphs with a given mean degree of vertices

Three-Dimensional Grid Drawings with Sub-quadratic Volume

“…The next generalisation of Theorem 7 for graphs with no K h -minor follows from Lemma 1(e), Theorem 5, and the result independently due to Kostochka [21] and Thomason [26] that χ(G) ∈ O(h log 1/2 h). 3/5 , which can be proved in a similar fashion to Lemma 4, in conjunction with the result of Fertin et al [15] that χ st (G) ≤ 20∆ 3/2 (see [11]).…”

On the Maximum Number of Cliques in a Graph

“…Kostochka [21] and Thomason [45] independently proved that G is O(t √ log t)-degenerate 2 . Thus Proposition 2 implies that G has at most 2 O(t √ log t) n cliques; similar bounds can be found in [28,32].…”

Number of Cliques in Graphs with a Forbidden Subdivision

“…The problem of determining the maximum number of edges in graphs with no K t -minor or no K t -subdivision is a well-studied problem in extremal graph theory: Kostochka [8] and Thomason [14] proved that graphs with no K t -minor have average degree at most ct √ ln t, and Bollobás and Thomason [1], and independently, Komlós and Szemerédi [7] proved that graphs with no K t -subdivision have average degree at most c ′ t 2 , where c and c ′ are some absolute constants not depending on t (in fact, a theorem of Thomas and Wollan [13] can be used to show that c ′ ≤ 10, see, [2, Theorem 7.2.1]). A graph is d-degenerate if all its induced subgraphs contain a vertex of degree at most d. The results mentioned above straightforwardly imply that graphs with no K t -minor are ct √ ln t-degenerate, and graphs with no K t -subdivision are c ′ t 2 -degenerate.…”