We provide a short and self-contained proof of the classical result Kostochka Thomason, ensuring that every graph average degree d $d$ has complete minor order Ω ( ∕ log ) ${\rm{\Omega }}(d\unicode{x02215}\sqrt{{\rm{log}}d})$ .

An independent transversal of a graph G $G$ with vertex partition P ${\mathscr{P}}$ is an set intersecting each block in single vertex. Wanless and Wood proved that if has size at least t $t$ the average degree vertices most ∕ 4 $t\unicode{x02215}4$ , then exists. We present construction showing this result optimal: for any ε > 0 $\varepsilon \gt 0$ sufficiently large there forest into parts su...

A graph G is called (k, 1)-colorable, if the vertex set of G can be partitioned into subsets V1 and V2 such that the graph G[V1] induced by the vertices of V1 has maximum degree at most k and the graph G[V2] induced by the vertices of V2 has maximum degree at most 1. We prove that every graph with a maximum average degree less than 10k+22 3k+9 admits a (k, 1)-coloring, where k ≥ 2. In particula...

max independent set is a paradigmatic problem in theoretical computer science and numerous studies tackle its resolution by exact algorithms with non-trivial worst-case complexity. The best such complexity is, to our knowledge, the O(1.1889) algorithm claimed by (J. M. Robson, Finding a maximum independent set in time O(2), Technical Report 1251-01, LaBRI, Université de Bordeaux I, 2001) in his...

abstract knowledge of the coordination of the agricultural activities in every region with the weather and climate condition of that area is necessary for any kind of agriculture activity. therefore, understanding the climate and analyzing the ecophysiological characteristics of plants are the most important factors in production. saffron is one of the most valuable plants, which is planted in ...