Download Lattice Path Matroids Are 3-Colorable

D ISKRETE M ATHEMATIK UND O PTIMIERUNG
Immanuel Albrecht and Winfried Hochstättler:
Lattice Path Matroids Are 3-Colorable
Technical Report feu-dmo039.15
Contact: immanuel.albrechtfernuni-hagen.de, [email protected]
FernUniversität in Hagen
Fakultät für Mathematik und Informatik
Lehrgebiet für Diskrete Mathematik und Optimierung
D – 58084 Hagen
2000 Mathematics Subject Classification: 05C15, 05B35, 52C40
Keywords: colorings, lattice path matroids, transversal matroids, oriented matroids
LATTICE PATH MATROIDS ARE 3-COLORABLE
IMMANUEL ALBRECHT AND WINFRIED HOCHSTÄTTLER
A BSTRACT. We show that every lattice path matroid of rank at least two has a positive
coline. As a corollary we get that any orientation of a lattice path matroid is 3-colorable.
Fakultät für Mathematik und Informatik, Fernuniversität in Hagen
Universitätsstr. 1, 58084 Hagen, Germany
[email protected]
[email protected]
1. I NTRODUCTION
Recently, in order to verify a generalized version of Hadwiger’s Conjecture to oriented
matroids for the case of 3-colorability, Goddyn et. al. [2] introduced the class of generalized
series parallel (GSP) matroids and asked whether it coincides with the class of oriented
matroids without M(K4 )-minor. Furthermore, they showed that a minor closed class C of
oriented matroids is a subclass of the GSP-matroids, if every simple matroid in C contains
a flat of codimension 2, i. e. a coline, which is contained in more flats of codimension
1, i. e. copoints, with only one extra element, than in larger copoints. We call such a
coline a positive coline. They conjectured that every simple gammoid of rank at least 2
has a positive coline. Gammoids are the smallest class of matroids which are closed under
minors and under duality that contain the transversal matroids, which are closed under
neither. Bicircular matroids form a minor closed class of the transversal matroids and
Goddyn et. al. [2] verified the existence of a positive coline in every simple representative
of rank at least 2 from that class.
Another minor closed subclass of the transversal matroids has recently attracted some
attention, the lattice path matroids [1]. In this note we show that every simple lattice path
matroids of rank at least 2 has a positive coline.
2. D EFINITIONS AND P REVIOUS R ESULTS
Definition 1. ([2], Def. 4) Let M = (E, rk, cl) be a matroid. A set X ⊆ E is called coline, if
X = cl(X) and rk(X) = rk(E) − 2. A set Y ⊆ E is called copoint on X, if X ⊆ Y , Y = cl(Y )
and rk(Y ) = rk(E) − 1. If further |Y \X| = 1, we say that Y is a simple copoint on X.
Otherwise, we say that Y is a fat copoint on X. A coline X ⊆ E is called positive, if there
are strictly more simple than fat copoints on X.
The following definitions are basically those found in [1].
Definition 2. Let n ∈ N. A lattice path of length n is a tuple (pi )ni=1 ∈ {N, E}[n] , and we
say that the i-th step is towards the North (East), if pi = N (pi = E).
1
LATTICE PATH MATROIDS ARE 3-COLORABLE
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Definition 3. Let n ∈ N and (pi )ni=1 , (qi )ni=1 be lattice paths of length n. We say that (pi )ni=1
is never above (qi )ni=1 with common endpoint if for all k ∈ [n],
|{i ∈ [k] | pi = N}| ≤ |{i ∈ [k] | qi = N}|
and |{i ∈ [n] | pi = N}| = |{i ∈ [n] | qi = N}| holds. In that case, we write (pi )ni=1 (qi )ni=1 .
Definition 4. Let (pi )ni=1 , (qi )ni=1 be lattice paths, such that (pi )ni=1 (qi )ni=1 . We define
the set of lattice paths between (pi )ni=1 and (qi )ni=1 to be
o
n
P[(pi )ni=1 , (qi )ni=1 ] = (ri )ni=1 ∈ {N, E}[n] | (pi )ni=1 (ri )ni=1 (qi )ni=1 .
Definition 5. A matroid M is called lattice path matroid, if it is isomorphic to the transversal matroid M[(pi )ni=1 , (qi )ni=1 ] presented by (Ak )rk=1 with
Ak = j ∈ [n] | ∃(ri )ni=1 ∈ P[(pi )ni=1 , (qi )ni=1 ] : r j = N ∧ k = |{i ∈ [ j] | ri = N}|
for some lattice paths (pi )ni=1 and (qi )ni=1 with (pi )ni=1 (qi )ni=1 and r = |{i ∈ [n] | pi = N}|.
Example.
Q
4
5
6
2
3
4
5
1
2
3
P
Let P = (E, E, N, E, N, N) and Q =
(N, N, E, N, E, E).
Then P Q and M[P, Q] is the transversal
matroid presented by (A1 , A2 , A3 ) where A1 =
{1, 2, 3}, A2 = {2, 3, 4, 5}, A3 = {4, 5, 6}.
Theorem 6. ([1], Thm. 2.1) Let P, Q be lattice paths, such that P Q. There is a bijection
between the bases of the lattice path matroid for P and Q, and the set of lattice paths
between P and Q, namely
∼
R 7→ R−1 (N)
P[P, Q] −→ B(M[P, Q]),
3. O UR RESULT
Given a representation P Q of a lattice path matroid there is a distinguished coline
containing the elements with smallest labels. In the following we will prove that, essentially, this coline is positive. We start with some simple observations.
Proposition 7. Let P = (pi )ni=1 , Q = (qi )ni=1 be lattice paths, such that P Q, let j ∈ [n].
With regard to M = M[P, Q],
(1) rk([ j]) = |{i ∈ [ j] | qi = N}|.
(2) j is a loop, if and only if
|{i ∈ [ j − 1] | pi = N}| = |{i ∈ [ j] | qi = N}| ,
i.e. the j-th step is forced to go towards East.
j
(3) for all k ∈ [n]\[ j], j and k are parallel edges, if and only if
=
|{i ∈ [ j − 1] | pi = N}| = |{i ∈ [k − 1] | pi = N}|
|{i ∈ [ j] | qi = N}| − 1 = |{i ∈ [k] | qi = N}| − 1
i.e. the j-th and k-th steps are in common a corridor towards the East that is one
step wide.
LATTICE PATH MATROIDS ARE 3-COLORABLE
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k
j
k
j
Proof.
(1) This follows from the fact ∀P R Q : |[ j] ∩ R−1 (N)| ≤ |[ j] ∩ Q−1 (N)|.
(2) j is a loop, if and only if rk({ j}) = 0, if and only if ∀B ∈ B(M) : j ∈
/ B, if
and only if ∀ ∈ (ri )ni=1 ∈ P[P, Q] : r j = E, if and only if |{i ∈ [ j − 1] | pi = N}| =
|{i ∈ [ j] | qi = N}|.
(3) “⇒”: Let j and k be parallel edges with j < k. Then ∀B ∈ B(M) : { j, k} 6⊆ B.
Thus
@P (ri )ni=1 Q : r j = N = rk (∗).
Since j is no loop, there is some (ri1 )ni=0 ∈ P[P, Q] with r1j = N which is minimal
with regard to . Then
i ∈ [ j − 1] | ri1 = N = |{i ∈ [ j − 1] | pi = N}| .
With (∗) follows
|{i ∈ [k] | qi = N}| = i ∈ [ j] | ri1 = N = i ∈ [ j − 1] | ri1 = N + 1.
Since k is no loop,
|{i ∈ [ j − 1] | pi = N}| = |{i ∈ [ j] | qi = N}| − 1.
“⇐”: Obviously, j and k appear only once and in the same Ai for 1 ≤ i ≤ r.
Lemma 8. Let P = (pi )ni=1 , Q = (qi )ni=1 be lattice paths, such that P Q and such that
M = M[P, Q] is loopless, let j ∈ [n] such that q j = N. Then
[ j − 1] = cl([ j − 1]).
Furthermore, if k ≥ j,
rk([ j − 1] ∪ {k}) = rk([ j − 1]) + 1.
Proof. By Proposition 7 (1), we get rk([ j − 1]) = |{i ∈ [ j − 1] | qi = N}|. Now fix some
k ∈ [n]\[ j − 1]. Since M has no loop, by Theorem 6 there is R = (ri )ni=0 ∈ P[P, Q] with
rk = N. We can construct a path S = (si )ni=0 ∈ P[P, Q] that follows Q for the first j − 1 steps,
then goes towards the East until it meets R, and then goes on as R does.
k
j
Q
R
l
LATTICE PATH MATROIDS ARE 3-COLORABLE
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The base S−1 (N) that corresponds to the constructed path yields
rk([ j − 1] ∪ {k}) ≥ ([ j − 1] ∪ {k}) ∩ S−1 (N) = 1 + [ j − 1] ∩ Q−1 (N) = 1 + rk([ j − 1]).
Since adding a single element to a set can at most add one to the rank, the inequality
in the above formula may be replaced by equality. This implies that k ∈
/ cl([ j − 1]) and
[ j − 1] = cl([ j − 1]).
Theorem 9. Let P = (pi )ni=1 , Q = (qi )ni=1 be lattice paths, such that P Q and such that
M = M[P, Q] is simple, let j1 = max(Q−1 (N)) and j2 = max(Q−1 (N)\{ j1 }). Then:
(1) [ j2 − 1] is a coline that we call the Western coline.
(2) [ j1 − 1] is a copoint on [ j2 − 1], which is fat if j1 − j2 ≥ 2,
(3) [ j2 − 1] ∪ {k} is a simple copoint on [ j2 − 1] for every k ≥ j1 .
Proof. That [ j2 − 1] is indeed a coline and [ j1 − 1] is indeed a copoint follows from
Lemma 8 and the construction of j2 and j1 . The equation |[ j1 − 1]\[ j2 − 1]| = j1 − j2 gives
the fatness property of [ j1 − 1]. Lemma 8 also provides that rk([ j2 − 1] ∪ {k}) = rk([n]) − 1.
The only thing left to show is that for any k ∈ [n] with k ≥ j1 , cl([ j2 − 1] ∪ {k}) =
[ j2 − 1] ∪ {k}, i.e. that those sets are simple copoints. Again, it suffices to construct a path
that shows that for any k0 ∈ [n]\([ j2 − 1] ∪ {k}), rk ([ j2 − 1] ∪ {k, k0 }) = rk([ j2 − 1]) + 2.
Possibly switching k and k0 , we may assume that k < k0 . Since M has neither parallel edges
nor loops, there is a lattice path R = (ri )ni=0 ∈ P[P, Q] such that {k, k0 } ⊆ R−1 (N).
j1
j2
Q
R
k0
k
< j1
≥ j1
Again, the path S = (si )ni=0 ∈ P[P, Q] follows Q for the first j2 − 1 steps, then goes towards the East until it meets R, and then goes on as R does. In this case, the
con0 }) ≥ ([ j − 1] ∪ {k, k0 }) ∩ S−1 (N) = 2 +
structed
path
S
yields
that
rk([
j
−
1]
∪
{k,
k
2
2
[ j2 − 1] ∩ Q−1 (N) = 2 + rk([ j2 − 1]).
Theorem 10. For every simple lattice path matroid M of rank ≥ 2, the Western coline is
positive, or n is a coloop and there is either another coloop or the rank is ≥ 3.
Proof. If j1 ≤ n − 1 as defined in Theorem 9, [ j2 − 1] has at most 1 fat and at least 2 simple
copoints, therefore it is positive. Otherwise j1 = n is a coloop. If there is another coloop
e1 , then [n − 1]\{e1 } is a positive coline with two simple copoints. If n is the only coloop,
rank 2 and no other coloop would imply that there are parallel edges.
Corollary 11. Every simple lattice path matroid M of rank ≥ 2 has a positive coline.
Proof. We proceed by induction on rk(M) ≥ 2. If rk(M) = 2, since M is simple, 0/ is a
positive coline. Assume rk(M) ≥ 3. Either the Western coline is a positive coline or n is
a coloop and M/n = M\n has a positive coline W by induction, and W ∪ {n} is a positive
coline of M.
Definition. ([2], Def. 2) Let O be a simple oriented matroid. We say that O is generalized
series-parallel, if every simple minor of O has a {0, ±1}-valued coflow which has exactly
one or two nonzero-entries.
LATTICE PATH MATROIDS ARE 3-COLORABLE
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Lemma. ([2], Lem. 5) If an orientable matroid M has a positive coline, then every orientation O of M has a {0, ±1}-valued coflow which has exactly one or two nonzero-entries.
Theorem. ([1], Thm. 3.1) The class of lattice path matroids is closed under minors, duals
and direct sums.
Corollary 12. All orientations of lattice path matroids are generalized series-parallel.
Theorem. ([2], Thm. 3) Every generalized-series parallel oriented matroid without loops
has a NZ-3 coflow, i.e. it is 3-colorable.
Corollary 13. Every loopless oriented lattice path matroid is 3-colorable.
R EFERENCES
[1] Joseph E. Bonin and Anna de Mier. Lattice path matroids: Structural properties. European Journal of Combinatorics, 27(5):701 – 738, 2006.
[2] Luis Goddyn, Winfried Hochstättler, and Nancy Ann Neudauer. Bicircular matroids are 3-colorable. Technical report, Discrete Mathematics and Optimization, FernUni in Hagen, 2015.