On the tree-depth and tree-width in heterogeneous random graphs

Abstract

In this note, we investigate the tree-depth and tree-width in a heterogeneous random graph obtained by including each edge $e_{ij}$ $(i\neq j)$ of a complete graph $K_{n}$ over $n$ vertices independently with probability $p_{n}(e_{ij})$. When the sequence of edge probabilities satisfies some density assumptions, we show both tree-depth and tree-width are of linear size with high probability. Moreover, we extend the method to random weighted graphs with non-identical edge weights and capture the conditions under which with high probability the weighted tree-depth is bounded by a constant.