Introduction to multimatroids and their polynomials Robert Brijder Hasselt University, Belgium Dagstuhl, June 12-17, 2016 Robert Brijder Multimatroid introduction
Motivation: 4-regular graphs v 2 e 1 t v , 1 = {{ e 1 , e 4 } , { e 2 , e 3 }} e 3 v 1 v 4 v t v , 2 = {{ e 1 , e 2 } , { e 3 , e 4 }} e 2 t v , 3 = {{ e 1 , e 3 } , { e 2 , e 4 }} e 4 v 3 G G | t v , 1 G | t v , 3 G | t v , 2 v ′′ v ′ v ′ v ′′ v ′ v ′′ Robert Brijder Multimatroid introduction
Motivation: 4-regular graphs Triple Z G = ( U , Ω , C ), where U = all transitions = { t v , i | v ∈ V ( G ) , i ∈ { 1 , 2 , 3 }} Ω = partition of U w.r.t. vertices = {{ t v , 1 , t v , 2 , t v , 3 } | v ∈ V ( G ) } C = minimal sets of (nonconflicting) transitions such that cutting along them increases the number of connected components v 2 C = {{ t v 4 , 2 } , { t v 1 , 3 , t v 2 , 3 } , { t v 1 , 3 , t v 3 , 3 } , v 1 v 4 { t v 2 , 3 , t v 3 , 3 } , { t v 1 , 2 , t v 2 , 1 , t v 3 , 1 }} v 3 G Robert Brijder Multimatroid introduction
Motivation: 4-regular graphs Triple Z G = ( U , Ω , C ), where U = all transitions = { t v , i | v ∈ V ( G ) , i ∈ { 1 , 2 , 3 }} Ω = partition of U w.r.t. vertices = {{ t v , 1 , t v , 2 , t v , 3 } | v ∈ V ( G ) } C = minimal sets of (nonconflicting) transitions such that cutting along them increases the number of connected components v 2 C = {{ t v 4 , 2 } , { t v 1 , 3 , t v 2 , 3 } , { t v 1 , 3 , t v 3 , 3 } , v 1 v ′ v ′′ 4 4 { t v 2 , 3 , t v 3 , 3 } , { t v 1 , 2 , t v 2 , 1 , t v 3 , 1 }} v 3 G Robert Brijder Multimatroid introduction
Motivation: 4-regular graphs Triple Z G = ( U , Ω , C ), where U = all transitions = { t v , i | v ∈ V ( G ) , i ∈ { 1 , 2 , 3 }} Ω = partition of U w.r.t. vertices = {{ t v , 1 , t v , 2 , t v , 3 } | v ∈ V ( G ) } C = minimal sets of (nonconflicting) transitions such that cutting along them increases the number of connected components v 2 C = {{ t v 4 , 2 } , { t v 1 , 3 , t v 2 , 3 } , { t v 1 , 3 , t v 3 , 3 } , v 1 v 4 { t v 2 , 3 , t v 3 , 3 } , { t v 1 , 2 , t v 2 , 1 , t v 3 , 1 }} v 3 G Robert Brijder Multimatroid introduction
Motivation: 4-regular graphs Triple Z G = ( U , Ω , C ), where U = all transitions = { t v , i | v ∈ V ( G ) , i ∈ { 1 , 2 , 3 }} Ω = partition of U w.r.t. vertices = {{ t v , 1 , t v , 2 , t v , 3 } | v ∈ V ( G ) } C = minimal sets of (nonconflicting) transitions such that cutting along them increases the number of connected components v ′′ 2 v ′ 2 v ′′ 1 C = {{ t v 4 , 2 } , { t v 1 , 3 , t v 2 , 3 } , { t v 1 , 3 , t v 3 , 3 } , v ′ v 4 1 { t v 2 , 3 , t v 3 , 3 } , { t v 1 , 2 , t v 2 , 1 , t v 3 , 1 }} v 3 G Robert Brijder Multimatroid introduction
Motivation: 4-regular graphs Triple Z G = ( U , Ω , C ), where U = all transitions = { t v , i | v ∈ V ( G ) , i ∈ { 1 , 2 , 3 }} Ω = partition of U w.r.t. vertices = {{ t v , 1 , t v , 2 , t v , 3 } | v ∈ V ( G ) } C = minimal sets of (nonconflicting) transitions such that cutting along them increases the number of connected components v 2 C = {{ t v 4 , 2 } , { t v 1 , 3 , t v 2 , 3 } , { t v 1 , 3 , t v 3 , 3 } , v 1 v 4 { t v 2 , 3 , t v 3 , 3 } , { t v 1 , 2 , t v 2 , 1 , t v 3 , 1 }} v 3 G Robert Brijder Multimatroid introduction
Motivation: 4-regular graphs Definition Let Ω be a partition of a finite set U . The elements ω ∈ Ω are called skew classes . A transversal T ⊆ U of Ω is such that | T ∩ ω | = 1 for all ω ∈ Ω. Theorem (Bouchet, 1997) For each transversal T, Z G [ T ] := ( T , C ∩ 2 T ) is a matroid described by its circuits. Example C = {{ t v 4 , 2 } , { t v 1 , 3 , t v 2 , 3 } , { t v 1 , 3 , t v 3 , 3 } , { t v 2 , 3 , t v 3 , 3 } , { t v 1 , 2 , t v 2 , 1 , t v 3 , 1 }} Transversal T = { t v 1 , 3 , t v 2 , 3 , t v 3 , 3 , t v 4 , 2 } . Z G [ T ] = ( T , {{ t v 4 , 2 } , { t v 1 , 3 , t v 2 , 3 } , { t v 1 , 3 , t v 3 , 3 } , { t v 2 , 3 , t v 3 , 3 }} ) is a matroid (isomorphic to U 1 , 3 ⊕ { loop } ). Robert Brijder Multimatroid introduction
Motivation: 4-regular graphs There is at most one way to cut a vertex to increase the number of connected components. v G | t v , 1 G | t v , 3 G | t v , 2 v ′′ v ′ v ′′ v ′ v ′′ v ′ NO SPLIT POSSIBLE SPLIT NO SPLIT Robert Brijder Multimatroid introduction
Definition multimatroid A subtransversal of Ω is a subset of a transversal of Ω. Definition (Bouchet, 1997) Let Ω be a partition of a finite set U , and let C be a set of subtransversals of Ω. Then Z = ( U , Ω , C ) is called a multimatroid if 1 for all transversals T , Z [ T ] := ( T , C ∩ 2 T ) is a matroid, and 2 for all C 1 , C 2 ∈ C , there are zero or at least two ω ∈ Ω with | ( C 1 ∪ C 2 ) ∩ ω | = 2. Called “multimatroid” because it contains multiple matroids (one for each transversal). Condition 2 formalizes that there is at most one way to cut a vertex to increase the number of connected components. Multimatroids can also be defined in terms of independent sets, rank, etc. Z G is a multimatroid, where | ω | = 3 for all ω ∈ Ω. Robert Brijder Multimatroid introduction
Minors Definition (minor operations) Let Z = ( U , Ω , C ) be a multimatroid and ω ∈ Ω. For u ∈ ω , we define Z | u = ( U \ ω, Ω \ { ω } , C ′ ) , where C ′ = � T C ( Z [ T ∪ u ] / u ) and T ranges over all transversals of Z | u . Theorem (Bouchet, 1998) Z | u is a multimatroid. Extendable to subtransversals S , denoted Z | S . Multimatroids of the form Z | S are called minors of Z . Theorem (Bouchet, 1998) For any 4-regular graph G and transition t of G, Z G | t = Z G | t . Robert Brijder Multimatroid introduction
Multimatroids and 4-regular graphs Theorem (Bouchet, 1997) Let G be a 4-regular graph. For all transversals T, n ( Z G [ T ]) = c ( G | T ) − c ( G ) , where n denotes the nullity of a matroid. The bases of Z G correspond 1-to-1 to Eulerian circuits of G. Robert Brijder Multimatroid introduction
Multimatroids from (delta-)matroids Definition Multimatroid Z is called a k-matroid if | ω | = k for all ω ∈ Ω. Example: Z G is a 3-matroid. A k -matroid has k kinds of minor operations. A 1-matroid corresponds to an ordinary matroid, but with only contraction (no deletion). A 2-matroid corresponds to a delta-matroid, with both deletion and contraction. So, a matroid can be viewed as a 1-matroid and a 2-matroid. Robert Brijder Multimatroid introduction
Tight multimatroids Definition (Bouchet, 2001) A multimatroid is tight if each minor with | Ω | = 1 has a circuit. Z G is a tight 3-matroid. Matroids form a subclass of tight 2-matroids. { t v , 2 } circuit of Z G ′ v G ′ = G | S G ′ | t v , 1 G ′ | t v , 3 G ′ | t v , 2 v ′′ v ′ v ′′ v ′ v ′′ v ′ NO SPLIT SPLIT! NO SPLIT Robert Brijder Multimatroid introduction
Tight 3-matroids Theorem Every k-matroid Z has (up-to-isomorphism) at most one tight ( k + 1) -matroid Z ′ with Z = Z ′ [ U \ T ] for some transversal T of Z ′ . So, some 2-matroids can also be viewed as tight 3-matroids. This includes all quaternary matroids. So, a quaternary matroid may be viewed as a 1-matroid, a 2-matroid, and a tight 3-matroid! Robert Brijder Multimatroid introduction
Multimatroid polynomial Definition ( B´ enard, Bouchet, Duchamp, 1997, B, Hoogeboom, 2014) Let Z = ( U , Ω , C ) be a multimatroid. We define the transition polynomial of Z as � y n ( Z [ T ]) , Q ( Z ; y ) = T where n ( Z [ T ]) denotes the nullity of matroid Z [ T ]. It turns out that (a multivariate version of) Q ( Z ; y ) subsumes various known polynomials, including: Tutte polynomial for matroids for part of the ( x , y )-plane, Martin polynomial for 4-regular graphs and 2-in, 2-out graphs, interlace polynomial for graphs, and Bollob´ as-Riordan polynomial (and others) for embedded graphs [Chun, Moffatt, Noble, Rueckriemen, 2014]. Robert Brijder Multimatroid introduction
Recursive relation Call u ∈ U singular if { u } ∈ C . Call ω ∈ Ω singular if ω contains a singular element. Theorem Let Z be a multimatroid. If Z is the empty multimatroid, then Q ( Z ; y ) = 1 . Otherwise let ω ∈ Ω . If ω is nonsingular in Z, then � Q ( Z ; y ) = Q ( Z | v ; y ) . v ∈ ω If ω is singular in Z, then for all w ∈ ω Q ( Z ; y ) = ( y + | ω | − 1) Q ( Z | w ; y ) , where u ∈ ω is singular in Z. Robert Brijder Multimatroid introduction
Evaluations Theorem Let Z be a tight k-matroid with k > 1 and U � = ∅ . Then Q ( Z ; 1 − k ) = 0 and For all transversals T, Q ( Z [ U \ T ]; 1 − k ) = ( − 1) | Ω | (1 − k ) n ( Z [ T ]) . [B, Hoogeboom, 2014] Tutte polynomial T ( M ; x , y ). Corollary Let M be a matroid. Case k = 2 . T ( M ; 0 , 0) = 0 and Case k = 3 . If M is quaternary, then T ( M ; − 1 , − 1) = ( − 1) | E ( M ) | ( − 2) n ( B ( M )) , where B ( M ) is the “bicycle matroid” of M. [Vertigan, 1998] Robert Brijder Multimatroid introduction
Open problem Open problem: a useful matroid-theoretic formalization of the Martin polynomial for Eulerian graphs in general. Including, e.g., 6-regular graphs. Problem: cutting a vertex of a 6-regular graph may increase the number of connected components by two . So, perhaps Z [ T ] should be a polymatroid instead of a matroid? In other words, do we need a “poly-multimatroid”? Or do we cut a degree 6 vertex first into degree 4 and degree 2? Robert Brijder Multimatroid introduction
Thanks! Robert Brijder Multimatroid introduction
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