Intersection Dimension and Graph Invariants

We show that the intersection dimension of graphs with respect to several hereditary properties can be bounded as a function of the maximum degree. As an interesting special case, we show that the circular dimension of a graph with maximum degree Δ is at most O(ΔlogΔlog logΔ)O\left( {\Delta {{\log \Delta } \over {\log \,\log \Delta }}} \right) . It is also shown that permutation dimension of any graph is at most Δ(log Δ)1+o(1). We also obtain bounds on intersection dimension in terms of treewidth