Tree representations of graphs

A graph is chordal if and only if it is the intersection graph of some family of subtrees of a tree. Applying "tolerance" allows larger families of graphs to be represented by subtrees. A graph G is in the family [Δ, d, t] if there is a tree with maximum degree Δ and subtrees corresponding to the vertices of G such that each subtree has maximum degree at most d and two vertices of G are adjacent if and only if the subtrees corresponding to them have at least t common vertices. It is known that both [3, 3, 1] and [3, 3, 2] are equal to the family of chordal graphs. Furthermore, one can easily observe that every graph G belongs to [3, 3, t] for some t. Denote by t (G) the minimum t so that G ∈ [3, 3, t]. In this paper, we study t (G) and parameters t (n) = min {t : G ∈ [3, 3, t] for every G ⊆ K_n} and t_bip (n) = min {t : G ∈ [3, 3, t] for every G ⊆ K_{n, n}} . In particular, our results imply that log n < t_bip (n) ≤ 5 n^{1 / 3} log₂ n and log (n / 2) < t (n) ≤ 20 n^{1 / 3} log₂ n.