Extremal connectivity for topological cliques in bipartite graphs

AbstractLet d(s) be the smallest number such that every graph of average degree >d(s) contains a subdivision of Ks. So far, the best known asymptotic bounds for d(s) are (1+o(1))9s2/64⩽d(s)⩽(1+o(1))s2/2. As observed by Łuczak, the lower bound is obtained by considering bipartite random graphs. Since with high probability the connectivity of these random graphs is about the same as their average degree, a connectivity of (1+o(1))9s2/64 is necessary to guarantee a subdivided Ks. Our main result shows that for bipartite graphs this gives the correct asymptotics. We also prove that in the non-bipartite case a connectivity of (1+o(1))s2/4 suffices to force a subdivision of Ks. Moreover, we slightly improve the constant in the upper bound for d(s) from 1/2 (which is due to Komlós and Szemerédi) to 10/23

The First Zagreb Index, Vertex-Connectivity, Minimum Degree And Independent Number in Graphs

Let G be a simple, undirected and connected graph. Defined by M1(G) and RMTI(G) the first Zagreb index and the reciprocal Schultz molecular topological index of G, respectively. In this paper, we determined the graphs with maximal M1 among all graphs having prescribed vertex-connectivity and minimum degree, vertex-connectivity and bipartition, vertex-connectivity and vertex-independent number, respectively. As applications, all maximal elements with respect to RMTI are also determined among the above mentioned graph families, respectively

Combinatorics

This is the report on the Oberwolfach workshop on Combinatorics, held 1–7 January 2006. Combinatorics is a branch of mathematics studying families of mainly, but not exclusively, finite or countable structures – discrete objects. The discrete objects considered in the workshop were graphs, set systems, discrete geometries, and matrices. The programme consisted of 15 invited lectures, 18 contributed talks, and a problem session focusing on recent developments in graph theory, coding theory, discrete geometry, extremal combinatorics, Ramsey theory, theoretical computer science, and probabilistic combinatorics