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

O0 The structure of 4-connected K 2, 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

O1 Minors of graphs We say H is a minor of G if – we can identify each vertex v of H with a connected subgraph C v in G ; – C u and C v are vertex-disjoint when u � = v ; – if uv is an edge of H , then there is some edge between C u and C v in G . K 2 , 4 We say G is H -minor-free if it does not have H as a minor.

O2 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 K 3 ,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 K 3 ,t -minor-free graph has a cycle of length at least α ( t ) n β ( β does not depend on t ). Easy: 2 -connected K 2 , 3 -minor-free ⇒ K 4 or outerplanar ⇒ hamiltonian. Chen, Sheppardson, Yu & Zang, 2006: 2 -connected K 2 ,t -minor-free graphs have a cycle of length at least n/t t − 1 . E, Marshall, Ozeki & Tsuchiya, 2016/2018: All 3 -connected K 2 , 4 -minor-free graphs are hamiltonian, and so are all 3 -connected planar K 2 , 5 -minor-free graphs. O’Connell, 2018+ / E, Gaslowitz, O’Connell & Royle, 2018+: All 3 -connected K 1 , 1 , 4 - minor-free graphs except K 3 , 4 are hamiltonian, and so are all 3 -connected planar K 1 , 1 , 5 -minor-free graphs except the Herschel graph.

O3 Edge bounds Note that K 2 ,t -minor-free graphs are very sparse: can bound m = number of edges. J.S. Myers, 2003: For t ≥ 10 29 , K 2 ,t -minor-free ⇒ m ≤ ( t + 1)( n − 1) / 2 . Chudnovsky, Reed & Seymour, 2011: This holds for all t ≥ 2 . Myers: For all t ≥ 2 , K 1 + ( kK t ) shows bound is tight for infinitely many n . Can improve if restrict connectivity: Ding, 2017+: 5 -connected K 2 ,t -minor-free ⇒ n ≤ n 5 ( t ) ⇒ m ≤ c 5 ( t ) . CR&S with Norin & Thomas: 3 -connected K 2 , 5 - minor-free ⇒ m ≤ 5 n/ 2 + c ( t ) . CR&S: There are 4 -connected K 2 ,t -minor-free graphs with m = 5 n/ 2 for all even n ≥ 6 : C n/ 2 [ K 2 ] .

O4 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 , K 2 ,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 K 2 ,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? K 2 , 4 -minor-free graphs do not have strips. So place to look is K 2 , 5 -minor-free graphs. Easiest to start with 4 -connected ones.

O5 Constructing 4-connected K 2,5 -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) C 2 n (join vertices at distance ≤ 2 in C n ), n ≥ 5 , or (b) the line graph of a cyclically- 4 -edge-connected (c 4 ec) cubic graph. C 2 14 Lemma: Line graphs of c 4 ec cubic graphs (except K 4 ) always have a K 2 , 5 -minor. To generate n -vertex 4 -connected K 2 , 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 K 2 , 5 minors, and throw in C 2 n . Works both for computer generation and as proof strategy. Special case, Gaslowitz, Marshall & Yepremyan, 2015: Every 4 -connected planar K 2 , 5 -minor-free graph is a squared even cycle.

O6 Characterization of 4-connected K 2,5 -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 b b I X a a b b a a b b Q ∆ ( XIX ∆∆∆ XQXXXQ ) a a b b Main theorem: For graphs on at least 9 vertices, the following are equivalent: (a) G is a 4 -connected K 2 , 5 -minor-free graph. (b) G is a Q -sequence graph. (c) G is a 4 -connected minor of C p [ K 2 ] for some p . Proof idea: Splitting a vertex always gives a Q -sequence graph or a K 2 , 5 minor.

O7 Counting 4-connected K 2,5 -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 K 2 , 5 -minor-free graphs, using P´ olya’s Theorem to take the symmetries into account. Theorem: Let g n be the number of n -vertex 4 -connected K 2 , 5 -minor-free graphs, up to isomorphism. The ordinary generating function for ( g n ) ∞ n =0 is g ( x ) = − 1 − x − 3 x 2 − 2 x 3 − 6 x 4 − 3 x 5 − 8 x 6 + 5 x 8 1 − x + 2 f ( x ) + f ( x 2 ) + f ( x ) 2 ∞ � � 1 φ ( k ) 1 � + + 2 k log 4 − 4 f ( x 2 ) 1 − f ( x k ) k =1 1 where φ is Euler’s totient function and f ( x ) = x 2 (1 + x 2 + 1 − x ) . Asymptotically, g n ∼ α n as n → ∞ , 2 n where α ≈ 1 . 85855898 is the largest real root of 1 − x + x 2 − 2 x 3 − x 4 + x 5 .

O8 4-connected K 2, t -minor-free graphs for general t Our result for K 2 , 5 -minor-free graphs gives a general result. • By Ding’s structure theorem, 4 -connected K 2 ,t -minor-free graphs are constructed by adding strips to a finite set of base graphs. Each strip S is a minor of P p [ K 2 ] for some p . • If we add a K 4 on the attachment vertices of strip S , the result S + is (a) still 4 -connected and (b) a minor of C p [ K 2 ] . By our result, S + is a Q -sequence graph. Corollary of main result: For every t , the strips in a 4 -connected K 2 ,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 K 2 ,t -minor-free graphs. Conjecture: The number of n -vertex 4 -connected K 2 ,t -minor-free graphs up to isomorphism is asymptotic to γ ( t ) n β ( t ) α n 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.

O9 Ongoing work (with Ryan Solava) Conjecture: The number of n -vertex 4 -connected K 2 ,t -minor-free graphs up to isomorphism is asymptotic to γ ( t ) n β ( t ) α n 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 K 2 ,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 K 2 , 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.

O10 Future work • Hamiltonicity question: We know that 3 -connected planar K 2 , 5 -minor-free graphs are hamiltonian. We have infinitely many counterexamples to show that this does not hold for 3 -connected planar K 2 , 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!