Section: New Results

Advances in Graph Theory

Participants : Aurore Alcolei, Rémi de Joannis de Verclos, François Pirot, Jean-Sébastien Sereni.

Keywords:

graph theory, graph coloring, extremal graph theory, chromatic number, two-mode data networks

Motivated by notions brought forward by sociology, we confirm a conjecture by Everett, Sinclair, and Dankelmann [Some Centrality results new and old, J. Math. Sociology 28 (2004), 215–227] regarding the problem of maximizing closeness centralization in two-mode data, where the number of data of each type is fixed. Intuitively, our result states that among all networks obtainable via two-mode data, the largest closeness is achieved by simply locally maximizing the closeness of a node. Mathematically, our study concerns bipartite graphs with fixed size bipartitions, and we show that the extremal configuration is a rooted tree of depth 2, where neighbors of the root have an equal or almost equal number of children [24].

Using recently introduced techniques based on entropy compression, we proved that the acyclic chromatic number of a graph with maximum degree $\Delta$ is less than $2.835{\Delta }^{4/3}+\Delta$. This improved the previous upper bound, which was $50{\Delta }^{4/3}$ (see [91] which is now published in Journal of Combinatorics, 7(4):725–737, 2016).