[Paper Review] On paths, stars and wyes in trees

Sébastien Bubeck, Katherine Edwards+2 more|arXiv (Cornell University)|Jan 8, 2016

Advanced Combinatorial Mathematics1 references3 citations

TL;DR

This paper advances the understanding of local profiles in trees by proving a tight linear bound $ Y \leq 9S + P + 6 $ on the number of 5-vertex wyes, paths, and stars in any tree, establishing that the segment between the 1-millipede and 2-millipede trees forms a face of the 5-profile convex set. Surprisingly, for $ d \geq 4 $, the segment $[d\text{-millipede}, (d+1)\text{-millipede}]$ is not a face, as demonstrated by constructing generalized millipede trees whose profiles lie outside the convex hull of standard millipede trees.

ABSTRACT

We further the study of local profiles of trees. Bubeck and Linial showed that the set of 5-profiles contains a certain polytope, namely the convex hull of d-millipedes, and they proved that the segment [0-millipede, 1-millipede] corresponds to a face of the set of 5-profiles. Our main result shows that the segment [1-millipede, 2-millipede] also corresponds to a face. Surprisingly we also show that for d > 3 the segment [d-millipede, (d+1)-millipede] is not a face of the set of 5-profiles. We do so by exhibiting new trees which are generalized millipedes with intriguing patterns for their degree sequence. The plot thickens, and the set of 5-profiles remains a mysterious convex set.

Motivation & Objective

• To refine the known linear bound on 5-vertex subtree counts in trees, particularly improving the earlier bound $ Y \leq 36S + P + 4 $.
• To determine whether segments between consecutive $ d $-millipede trees form faces of the convex set $ \Delta_{\mathcal{T}}(5) $, the set of attainable 5-profiles in large trees.
• To resolve Open Problem 1 in Bubeck and Linial (2013) by characterizing the exact face corresponding to the new bound.
• To answer Open Problem 3 by showing that if the proportion of paths tends to zero, the proportion of stars tends to one, providing a non-linear relation between path and star frequencies.

Proposed method

• Uses inductive arguments based on a novel extension of the $(i,j)$-cut operation, generalizing the $(1,0)$-cuts used in prior work.
• Introduces $(0,0)$, $(1,0)$, $(1,1)$, and $(2,1)$-cuts to manipulate tree structures while tracking changes in $ P $, $ S $, and $ Y $.
• Employs a reduction lemma to eliminate degree-2 vertices without affecting the inequality, simplifying analysis to trees with minimum degree 3.
• Applies a degree-sequence-based upper bound on $ Y(T) $, showing $ Y(T) \leq 2\sum_v d_v^{2+\sqrt{3}} $, with $ 2+\sqrt{3} $ as the optimal exponent.
• Uses Jensen’s inequality and combinatorial bounds on subtree counts to derive non-linear inequalities relating path and star proportions in large trees.
• Constructs generalized millipede trees with specific degree sequences to demonstrate that $[d\text{-millipede}, (d+1)\text{-millipede}]$ is not a face for $ d \geq 4 $.

Experimental results

Research questions

• RQ1 Does the segment $[1\text{-millipede}, 2\text{-millipede}]$ correspond to a face of the 5-profile convex set $ \Delta_{\mathcal{T}}(5) $?
• RQ2 For $ d \geq 4 $, is the segment $[d\text{-millipede}, (d+1)\text{-millipede}]$ a face of $ \Delta_{\mathcal{T}}(5) $?
• RQ3 Can a tighter linear bound than $ Y \leq 36S + P + 4 $ be established for all trees?
• RQ4 What is the asymptotic behavior of the proportion of stars when the proportion of paths among 5-vertex subtrees tends to zero?
• RQ5 What is the optimal exponent in the degree-sequence bound for $ Y(T) $, and can it be achieved?

Key findings

• The bound $ Y \leq 9S + P + 6 $ is proven to be tight and optimal, with equality achievable for arbitrarily large $ S, P, Y $.
• The segment $[1\text{-millipede}, 2\text{-millipede}]$ corresponds to a face of the 5-profile convex set $ \Delta_{\mathcal{T}}(5) $, characterizing the new bound.
• For $ d \geq 4 $, the segment $[d\text{-millipede}, (d+1)\text{-millipede}]$ is not a face of $ \Delta_{\mathcal{T}}(5) $, as shown by constructing generalized millipede trees outside the convex hull of standard millipede trees.
• A new upper bound on $ Y(T) $ is established: $ Y(T) \leq 2\sum_v d_v^{2+\sqrt{3}} $, and the exponent $ 2+\sqrt{3} $ is proven optimal.
• A non-linear inequality is derived: $ p_2 \geq 1 - e(k-1)!(k-1)^\varepsilon p_1^{1-\varepsilon/(k-1)} $ for $ k \geq 6 $, with a specific form for $ k=5 $, resolving Open Problem 3.
• The result confirms that if the proportion of $ k $-vertex paths tends to zero, then the proportion of $ k $-vertex stars tends to one, with an explicit quantitative rate.