[Paper Review] Extremal density for sparse minors and subdivisions

John Haslegrave, Jaehoon Kim+1 more|arXiv (Cornell University)|Dec 3, 2020

Advanced Graph Theory Research34 references4 citations

TL;DR

This paper establishes asymptotically tight bounds on the average degree required to force subdivisions of sparse, bounded-degree bipartite graphs with a mild separability condition. Using structural graph theory and probabilistic methods, it proves that average degree $ (1+o(1))t^2 $ suffices for the $ t imes t $ grid as a topological minor, and $ (3/2+o(1))t $ forces all $ t $-vertex planar graphs as minors, with the constant $ 3/2 $ being optimal and universal across all fixed surfaces.

ABSTRACT

We prove an asymptotically tight bound on the extremal density guaranteeing subdivisions of bounded-degree bipartite graphs with a mild separability condition. As corollaries, we answer several questions of Reed and Wood on embedding sparse minors. Among others, $\bullet$ $(1+o(1))t^2$ average degree is sufficient to force the $t imes t$ grid as a topological minor; $\bullet$ $(3/2+o(1))t$ average degree forces every $t$-vertex planar graph as a minor, and the constant $3/2$ is optimal, furthermore, surprisingly, the value is the same for $t$-vertex graphs embeddable on any fixed surface; $\bullet$ a universal bound of $(2+o(1))t$ on average degree forcing every $t$-vertex graph in any nontrivial minor-closed family as a minor, and the constant 2 is best possible by considering graphs with given treewidth.

Motivation & Objective

• To determine the exact asymptotic average degree threshold that guarantees the existence of subdivisions of sparse, bounded-degree bipartite graphs with mild separability.
• To resolve open questions by Reed and Wood on embedding sparse minors, particularly concerning the $ t imes t $ grid and planar graphs.
• To establish a universal constant for forcing all $ t $-vertex graphs in any nontrivial minor-closed family as minors, with optimality analysis.
• To unify and extend previous results on extremal density for topological minors and minors in sparse graph families.

Proposed method

• Introduces the concept of $ eta $-separability for bipartite graphs, ensuring that vertex removals can limit component sizes, enabling inductive construction of subdivisions.
• Applies probabilistic and extremal graph techniques to show that a graph with average degree $ d $ contains a subdivision of any $ eta $-separable $ H $ with $ |H| o (1-eta)d $ and $ igtriangleup(H) o igtriangleup $.
• Uses the structure of planar and surface-embedded graphs to derive tight bounds via bipartite subdivision constructions and face-division arguments.
• Applies known results on chromatic number and minor-closed families to derive bounds of the form $ d_{ ext{min}} = (2 - 2/ ho + o(1))t $ for minor-closed families with chromatic number $ ho $.
• Leverages known extremal results on $ K_t $-minors and topological minors to calibrate the leading constants in the asymptotic bounds.
• Employs double-counting arguments on face cycles and 1-factor decompositions to prove optimality of subdivision size bounds in planar graphs.

Experimental results

Research questions

• RQ1 What is the minimal average degree that forces the $ t imes t $ grid as a topological minor?
• RQ2 Is the constant $ 3/2 $ in the average degree bound for forcing all $ t $-vertex planar graphs as minors optimal, and does it extend to graphs embeddable on any fixed surface?
• RQ3 Can a universal constant $ c $ be established such that $ ct $ average degree forces all $ t $-vertex graphs in any nontrivial minor-closed family as minors, and is $ c = 2 $ best possible?
• RQ4 For which graphs $ G $ does $ |2\alpha(G) - \alpha_2(G)| = o(|G|) $, and how does this relate to extremal minor density?
• RQ5 Does the bound $ d_{ ext{min}} = (2 - 2/\chi(\mathcal{F}) + o(1))t $ hold for all minor-closed families $ \mathcal{F} $, especially those not closed under disjoint union?

Key findings

• An average degree of $ (1+o(1))t^2 $ is sufficient and necessary to force the $ t imes t $ grid as a topological minor.
• An average degree of $ (3/2+o(1))t $ forces every $ t $-vertex planar graph as a minor, and the constant $ 3/2 $ is optimal, with the same bound applying to graphs embeddable on any fixed surface.
• A universal bound of $ (2+o(1))t $ average degree forces all $ t $-vertex graphs in any nontrivial minor-closed family as minors, and the constant 2 is best possible, as shown by treewidth-based extremal constructions.
• For any $ eta $-separable bipartite graph $ H $ with $ |H| o (1-eta)d $ and $ igtriangleup(H) o igtriangleup $, average degree $ d $ in $ G $ guarantees a subdivision of $ H $, with the bound being asymptotically tight.
• The bound $ d_{ ext{min}} = (2 - 2/\chi(\mathcal{F}) + o(1))t $ holds for minor-closed families $ \mathcal{F} $, and is tight for families closed under disjoint union.
• The class of linklessly embeddable graphs satisfies $ d_{ ext{min}} = (8/5 + o(1))t $, demonstrating a strictly smaller constant than the general $ 2t $ bound.