Geodesic
Intersection Dimension and Graph Invariants
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 1, pp. 153-166

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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(ΔlogΔ/log logΔ). 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.
Keywords: circular dimension, dimensional properties, forbidden-subgraph colorings 
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Aravind, N.R.; Subramanian, C.R. Intersection Dimension and Graph Invariants. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 1, pp. 153-166. http://geodesic.mathdoc.fr/item/DMGT_2021_41_1_a9/