The structure of 4-connected K 2, t -minor-free graphs Mark - - PowerPoint PPT Presentation

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The structure of 4-connected K2,t-minor-free graphs

Mark Ellingham Vanderbilt University

• J. Zachary Gaslowitz

The Proof School

Supported by NSA grant H98230-13-1-0233 and Simons Foundation award 429625

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Minors of graphs

We say H is a minor of G if – we can identify each vertex v of H with a connected subgraph Cv in G; – Cu and Cv are vertex-disjoint when u = v; – if uv is an edge of H, then there is some edge between Cu and Cv in G. K2,4 We say G is H-minor-free if it does not have H as a minor.

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Why exclude complete bipartite minors?

Excluding complete bipartite minors is similar to requiring toughness. Gives long cycles, spanning subgraphs of low degree, e.g.: Ota & Ozeki, 2012: A 3-connected K3,t-minor-free graph has a spanning tree of maximum degree at most t − 1 if t is even, and at most t if t is odd (sharp). Chen, Yu & Zang, 2012: A 3-connected n-vertex K3,t-minor-free graph has a cycle

• f length at least α(t)nβ

(β does not depend on t). Easy: 2-connected K2,3-minor-free ⇒ K4 or outerplanar ⇒ hamiltonian. Chen, Sheppardson, Yu & Zang, 2006: 2-connected K2,t-minor-free graphs have a cycle of length at least n/tt−1. E, Marshall, Ozeki & Tsuchiya, 2016/2018: All 3-connected K2,4-minor-free graphs are hamiltonian, and so are all 3-connected planar K2,5-minor-free graphs. O’Connell, 2018+ / E, Gaslowitz, O’Connell & Royle, 2018+: All 3-connected K1,1,4- minor-free graphs except K3,4 are hamiltonian, and so are all 3-connected planar K1,1,5-minor-free graphs except the Herschel graph.

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Edge bounds

Note that K2,t-minor-free graphs are very sparse: can bound m = number of edges. J.S. Myers, 2003: For t ≥ 1029, K2,t-minor-free ⇒ m ≤ (t + 1)(n − 1)/2. Chudnovsky, Reed & Seymour, 2011: This holds for all t ≥ 2. Myers: For all t ≥ 2, K1 + (kKt) shows bound is tight for infinitely many n. Can improve if restrict connectivity: Ding, 2017+: 5-connected K2,t-minor-free ⇒ n ≤ n5(t) ⇒ m ≤ c5(t). CR&S with Norin & Thomas: 3-connected K2,5- minor-free ⇒ m ≤ 5n/2 + c(t). CR&S: There are 4-connected K2,t-minor-free graphs with m = 5n/2 for all even n ≥ 6: Cn/2[K2].

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Ding’s structure theorem

P = outerplanar graphs where each chord can cross at most one other. Strip has two paths with chords between them where each chord can cross at most one other. Ding, 2017+: Loosely, for a given t, K2,t-minor-free graphs are built from P, together with base graphs of bounded order with disjoint strips and fans attached, by a bounded number of ‘2-sums’ (not usual definition). For 3-connected, just base graphs with added fans and strips. For 4-connected, just base graphs with added strips. Questions: Ding’s result is a necessary condition for a K2,t-minor-free graph. Can we get a necessary and sufficient condition? Also, can we provide more information regarding the structure of the strips in Ding’s result? K2,4-minor-free graphs do not have strips. So place to look is K2,5-minor-free

• graphs. Easiest to start with 4-connected ones.

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Constructing 4-connected K2,5-minor-free graphs

Martinov, 1982: Every 4-connected graph G has an edge e so that G/e (G contract e) is 4-connected, unless G is (a) C2

n (join vertices at distance ≤ 2 in Cn), n ≥ 5, or

(b) the line graph of a cyclically-4-edge-connected (c4ec) cubic graph. C2

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Lemma: Line graphs of c4ec cubic graphs (except K4) always have a K2,5-minor. To generate n-vertex 4-connected K2,5-minor-free graphs: Take those on n − 1 vertices, split one vertex (uncontract an edge) in all possible ways that preserve 4-connectivity, discard those with K2,5 minors, and throw in C2

n.

Works both for computer generation and as proof strategy. Special case, Gaslowitz, Marshall & Yepremyan, 2015: Every 4-connected planar K2,5-minor-free graph is a squared even cycle.

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Characterization of 4-connected K2,5-minor-free graphs

Q-sequence graphs: Obtained by gluing together a cyclic sequence of I, X, ∆ and Q subgraphs subject to: every I or Q must be surrounded by two X’s, and two consecutive ∆’s must face in opposite directions. a a a a a a a a b b b b b b b b I X ∆ Q (XIX∆∆∆XQXXXQ) Main theorem: For graphs on at least 9 vertices, the following are equivalent: (a) G is a 4-connected K2,5-minor-free graph. (b) G is a Q-sequence graph. (c) G is a 4-connected minor of Cp[K2] for some p. Proof idea: Splitting a vertex always gives a Q-sequence graph or a K2,5 minor.

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Counting 4-connected K2,5-minor-free graphs

Each sequence of I, X, ∆, Q (up to cyclic shifts and reversals) generates a unique

• graph. So we can count isomorphism classes of 4-connected K2,5-minor-free

graphs, using P´

• lya’s Theorem to take the symmetries into account.

Theorem: Let gn be the number of n-vertex 4-connected K2,5-minor-free graphs, up to isomorphism. The ordinary generating function for (gn)∞

n=0 is

g(x) = −1 − x − 3x2 − 2x3 − 6x4 − 3x5 − 8x6 + 5x8 + 1 1 − x + 2f(x) + f(x2) + f(x)2 4 − 4f(x2) +

∞

• k=1

φ(k) 2k log

• 1

1 − f(xk)

• where φ is Euler’s totient function and f(x) = x2(1 + x2 +

1 1 − x). Asymptotically, gn ∼ αn 2n as n → ∞, where α ≈ 1.85855898 is the largest real root of 1 − x + x2 − 2x3 − x4 + x5.

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4-connected K2,t -minor-free graphs for general t

Our result for K2,5-minor-free graphs gives a general result.

• By Ding’s structure theorem, 4-connected K2,t-minor-free graphs are constructed

by adding strips to a finite set of base graphs. Each strip S is a minor of Pp[K2] for some p.

• If we add a K4 on the attachment vertices of strip S, the result S+ is (a) still

4-connected and (b) a minor of Cp[K2]. By our result, S+ is a Q-sequence graph. Corollary of main result: For every t, the strips in a 4-connected K2,t-minor-free graph are (linear) Q-sequence graphs. This suggests that we should be able to get counting results (at least asymptotically) for 4-connected K2,t-minor-free graphs. Conjecture: The number of n-vertex 4-connected K2,t-minor-free graphs up to isomorphism is asymptotic to γ(t)nβ(t)αn as n → ∞, where α ≈ 1.85855898 as before, β(t) is an integer depending on the maximum number of ‘unrestricted’ strips that can occur, and γ(t) depends on the base graphs and how strips can connect to them.

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Ongoing work (with Ryan Solava)

Conjecture: The number of n-vertex 4-connected K2,t-minor-free graphs up to isomorphism is asymptotic to γ(t)nβ(t)αn as n → ∞, where α ≈ 1.85855898 as before, β(t) is an integer depending on the maximum number of ‘unrestricted’ strips that can occur, and γ(t) depends on the base graphs and how strips can connect to them.

• Work towards proving conjecture on number of n-vertex 4-connected K2,t-minor-

free graphs:

• Some technical issues need to be dealt with.
• Need to get good upper and lower bounds on number of strips we can have.
• Characterize 3-connected K2,5-minor-free graphs. We get a family of A-sequence

graphs (generalize Q-sequence graphs) plus a family where we add fans to about a thousand base graphs. Would like to use this structure to also obtain (asymptotic) counting results here. Main contribution to variability of graph comes from strips, not fans.

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Future work

• Hamiltonicity question: We know that 3-connected planar K2,5-minor-free graphs

are hamiltonian. We have infinitely many counterexamples to show that this does not hold for 3-connected planar K2,6-minor-free graphs. But computer results (Gordon Royle) suggest that the nonhamiltonian ones all fall into a simple family, so can we prove this? Thank you!