On Noneven Digraphs and Symplectic Pairs

Home , Ij (digraph)

Chjan C Lim

David A Schmidt

Department of Mathematics

Rensselaer Polytechnic Institute

June

Abstract

A digraph D is called noneven if it is p ossible to assign weights of to its arcs so that

D contains no cycle of even weight A noneven digraph D corresp onds to one or more

nonsingular sign patterns

Given an n n sign pattern H a symplectic pair in QH is a pair of matrices A D

such that A QH D QH and A D I An unweighted digraph D allows a matrix

prop erty P if at least one of the sign patterns whose digraph is D allows P

In Thomassen characterized the noneven connected symmetric digraphs ie di

graphs for which the existence of arc u v implies the existence of arc v u In the rst part

of our pap er we recall this characterization and use it to determine which strong symmetric

digraphs allow symplectic pairs

A digraph D is called semicomplete if for each pair of distinct vertices u v at

least one of the arcs u v and v u exists in D Thomassen again in presented a

characterization of two classes of strong noneven digraphs the semicomplete class and the

class of digraphs for which each vertex has total degree which exceeds or equals the size of

the digraph In the second part of our pap er we ll a gap in these two characterizations

and present and prove correct versions of the main theorems involved We then pro ceed to

determine which digraphs from these classes allow symplectic pairs

Introduction

A sign pattern is an m n matrix H whose entries are or We say that an m n

matrix A QH the signclass of H if sg nA sg nH for all pairs of indices i j

ij ij

where

sg nx

An n n sign pattern H is called signnonsingular or sometimes an LMatrix if for

all matrices A QH detA H is combinatorially singular if detA for all

A QH Signnonsingular matrices have b een studied extensively in the context of the

signsolvability problem

The weighted digraph of an n n sign pattern H is the directed graph D H with

n vertices and the arc set fi j i j H g with each arc assigned the weight if

H and the weight otherwise Since the diagonal entries of H are not reected in

D H it is convenient when p ossible to restrict consideration to negativediagonal sign

patterns ie sign patterns H for which H for all indices i

Our digraph terminology follows that of A digraph D is strongly connected or

strong if for any ordered pair of vertices u v from D there is a directed path b eginning

at u and terminating at v D is strongly k connected if the removal of k or fewer

vertices from D results in a strong digraph Vertex u dominates vertex v in D if the arc

u v exists in D The indegree of vertex v idv is the number of vertices that dominate

v and the outdegree of v o dv is the number of vertices dominated by v The degree

of v dv is the sum of v s indegree and outdegree Let D b e a directed graph of size

n with vertex set V A nite sequence fd d d g is an indegree sequence for D if

1 2 n

there is an ordering fv g i n of V such that idv d for i n An

i i i

outdegree sequence for D is dened analogously D is called semicomplete if for any

pair of vertices u v in D either u dominates v or vice versa The undirected cycle of length

k is denoted C with C denoting a single edge joining vertices

k 2

A weighted digraph D is called even if some directed cycle in D has even total weight

where the weight of a directed path or cycle in D is obtained by summing the weights of

the arcs contained in that path or cycle If D is not even it is called noneven A negative

diagonal signnonsingular matrix has the prop erty that its weighted digraph is noneven

An unweighted digraph D is called even if D contains an even cycle whenever the arcs

of D are assigned weights of or If there is at least one weight assignment for the arcs of

D that results in a noneven weighted digraph D is called noneven In view of the ab ove

it follows that any noneven digraph D corresp onds to at least one and p erhaps several

negativediagonal signnonsingular patterns

A digraph D is called symmetric if the existence of arc u v in D implies the existence

of the arc v u A negativediagonal sign pattern whose digraph is symmetric is called

combinatorially symmetric Every undirected graph G gives rise to a corresp onding

symmetric digraph which we shall call G An undirected graph G will b e called noneven if

G is noneven

A sub division of an arc u v in a digraph D is obtained by removing u v adding

a new vertex w and adding the arcs u w and w v To split a vertex v in D means

to replace v with two vertices v and v and an arc v v such that all arcs dominating

1 2 1 2

resp ectively dominated by v in D dominate v resp ectively are dominated by v in the

1 2

resulting digraph The result of splitting vertices and sub dividing arcs in the double cycle

is called a weak k doublecycle Seymour and Thomassen showed that a digraph C

D is even if and only if D contains a weak k doublecycle for some o dd k

Using terminology from let P b e a prop erty that a real square matrix may or may not

p ossess The sign pattern H is said to allow prop erty P if there exists a matrix A QH

such that A has prop erty P H is said to require P if all matrices A in QH have prop erty

P Two sign patterns G and H are signequivalent written G H if G can b e obtained

from H by negating some of its rows and columns and then p ermuting the rows and columns

Given two negativediagonal sign patterns G and H we will call their weighted di

graphs D G and D H signequivalent if G H A matrix prop erty P is called a sign

equivalence class prop erty if a sign pattern H allowing P implies that P is allowed by

all sign patterns that are signequivalent to H The problem of characterizing the patterns

allowing such a prop erty is simplied b ecause the analysis can b e carried out using only

negativediagonal sign patterns since any pattern that is not combinatorially singular is

signequivalent to some negativediagonal pattern

A weighted digraph D is said to allow matrix prop erty P if the negativediagonal sign

pattern corresp onding to D allows P An unweighted digraph D is said to allow P if some

negativediagonal sign pattern whose digraph is D allows P

A result of Brualdi and Shader states that any two negativediagonal nonsingular

sign patterns with the same unweighted noneven digraph D are signequivalent This result

implies that two weighted noneven digraphs D and D are signequivalent if and only their

1 2

indegree and outdegree sequences coincide It therefore allows us to classify completely

the negativediagonal signnonsingular patterns allowing a signequivalence class prop erty P

using only unweighted digraphs

Given an n n sign pattern H a symplectic pair in QH is a pair of matrices

A D such that A QH D QH and A D I Symplectic pairs are a pattern

generalization of orthogonal matrices which arise from a sp ecial symplectic matrix found in

nb o dy problems in celestial mechanics and have b een discussed previously in In

particular it was shown in by the authors that the symplectic pair prop erty is a sign

equivalence class prop erty In view of this fact the characterizations of signnonsingular

patterns allowing symplectic pairs discussed in this pap er are carried out in terms of un

weighted digraphs It should b e understo o d that if a noneven digraph D is shown to allow

symplectic pairs it follows that any negativediagonal signnonsingular pattern whose di

graph is D and any sign pattern that is signequivalent to such a pattern also allows

symplectic pairs

Overlap Numbers Digraphs and Symplectic Pairs

We rst discuss a necessary condition for a sign pattern to allow symplectic pairs Let H

b e an n n sign pattern Given disctinct indices i and j dene the signed row over

lap numbers N H resp ectively N H as the number of column indices k for which

ij ij

H H resp ectively The following necessary condition for a sign pattern H to

ik j k

allow symplectic pairs follows directly from the requirement that the sign pattern H p ermit

orthogonality of row vectors for A QH

Prop osition Let H b e an n n sign pattern If N H N H for some pair of

ij ij

distinct indices and H allows symplectic pairs then N H N H

ij ij

We will need a digraphical version of this necessary condition Assuming for the moment

that a vertex in the digraph D H dominates itself if H Prop osition can b e restated

as follows

Prop osition digraphical version Let D H b e the weighted digraph of the sign

pattern H Given a pair of distinct vertices v and w from D H let U b e the set of all

v w

vertices from D H which are dominated by b oth v and w Then U is the disjoint union

v w

of the two sets U and U the rst consisting of the vertices u for which the sum of the

v w v w

weights of the arcs v u and w u is even and the second consisting of those for which

this sum is o dd If there exists a pair of distinct vertices v and w for which exactly one of

the sets U and U is empty D H do es not allow symplectic pairs

v w v w

Figure shows a negativediagonal sign pattern H with its weighted digraph D H

which fails to allow symplectic pairs by Prop osition Note that for a negativediagonal

pattern H the selfarcs of D H inferred here all have unit weight

C B

C B H

10 2

1 1 1 0

4 0 3

D(H)

Figure H and D H Where H Fails to Allow Symplectic Pairs

Symmetric Noneven Digraphs and Symplectic Pairs

A digraph D is called symmetric if the existence of arc u v in D implies the existence of the

arc v u A negativediagonal matrix whose digraph is symmetric is called combinatorially

symmetric Every undirected graph G gives rise to a corresp onding symmetric digraph

which we shall call G An undirected graph G will b e called noneven if G is noneven

In Thomassen characterized the connected noneven symmetric digraphs Construct

the graph G as follows Starting with C choose any edge u v and add the vertices u

0 0 0 0 0

and v and add the edges u u u v and v v Rep eat this pro cedure as desired The

resulting connected graph G is called a C co ckade Thomassen showed that a connected

graph G is noneven if and only if G is a subgraph of a C co ckade and also proved a useful

equivalent condition which was proven indep endently by Harary et al

Prop osition The connected graph G is even if and only if G C n o dd or G

contains two vertices joined by three internally disjoint paths one of which has even length

We shall call a cycle C in a graph G reduced if the subgraph of G induced by the vertices

from C is C itself The following facts are direct consequences of Prop osition If G is a

connected subgraph of a C co ckade G must contain at least one reduced cycle of length

or greater In particular a connected subgraph G of a C co ckade is itself a C co ckade

4 4

if and only if all of its reduced cycles have length

We now consider the problem of determining which of the strong symmetric noneven

digraphs allow symplectic pairs

Theorem Let G b e a strong symmetric noneven digraph with underlying undirected

or G C graph G If G allows symplectic pairs then either G C

2 4

Pro of If G C then the signnonsingular pattern

C B

whose unweighted digraph is G allows indeed requires symplectic pairs see Chapter

Three If G C then the negativediagonal signnonsingular pattern

C B

whose digraph if G allows symplectic pairs and also allows the more restrictive prop erty of

orthogonality see

If G has connectivity one then G has a cut vertex nor C Now supp ose G is neither C

4 2

v and there exist vertices u and w distinct from v in G such that every uw path contains

v Consider an arbitrary uw path P at least one of which must exist since G is connected

0 0 0

and call the unique pair of vertices that are adjacent to v in P u and w In G b oth u and

w dominate v but cannot b oth dominate any other vertex of G since this would result in a

uw path in G not containing v Thus G fails to allow symplectic pairs by Prop osition

If G is connected G must b e a subgraph of a C co ckade Supp ose rst that G is a

C co ckade Since G C G must have the form shown in Figure where the subgraph G

4 4

of G is also a C co ckade

0 00

Note that u and u b oth dominate u in G and that the only other vertices dominated _ u' G u u" v'

v" v

Figure A C Co ckade Prop erly Containing C

4 4

0 0 00 0 0

by u in G are itself and v But u dominates neither u nor v Thus G fails to allow

symplectic pairs by Prop osition

Now supp ose that G is not a C co ckade which means that G contains a reduced cycle

C of length p Consider three vertices u v and w from C as shown in Figure such

that u and v b oth dominate w in G Since C is a reduced cycle u and v cannot b oth b e

adjacent to any other vertex in C since otherwise C would have length four Supp ose u and

v are b oth adjacent to a vertex x not in C as shown in Figure Then there exist three

internally disjoint paths joining u and v namely the two paths from u to v on C and the

path uxv Furthermore this last path has length two Thus by Prop osition G is even

which is a contradiction Thus no such vertex x exists in G and G fails to allow symplectic

pairs by Prop osition

Note that a symmetric digraph G C such that G has connectivity one fails to allow

symplectic pairs whether G is noneven or not u w C

Figure A Reduced Cycle of Length p

Dense Noneven Digraphs

We shall call a digraph D dense if D has one of the prop erties studied by Thomassen

with regard to the even cycle problem in ie either D is semicomplete or D is such

that the total degree of each vertex exceeds or equals the size of D In the study of dense

noneven digraphs an imp ortant role is played by the extended caterpillars An extended

caterpillar is a semicomplete digraph constructed as follows Consider a simple undirected

path P x x x With each vertex x j k asso ciate a p ossibly empty set

1 2 k j

of vertices fx g all of which are adjacent to x Call the resulting tree T Call the path P

j j

the backbone of T and call a particular backbone vertex x together with its asso ciated

g if any a blossom of T Note that the ordering of the backbone vertices of T vertices fx

induces through blossom membership an ordering on all the vertices of T in particular we

will write u v if u is in the blossom asso ciated with backbone vertex x v is in the blossom

asso ciated with backbone vertex x and i j We now form the extended caterpillar E by

adding to T all arcs u v not present in T such that u v and exactly one arc joining

any two nonbackbone vertices in a given blossom so that each blossom contains a transitive

tournament on its vertex set It is clear that any extended caterpillar is dense in b oth of the

senses discussed ab ove and it is easy to show that an extended caterpillar is noneven An

example of an extended caterpillar of size six is shown in Figure

Extended caterpillars and their strong sub digraphs have connectivity one Also a strong

sub digraph of an extended caterpillar E must contain a backbone path of the form x x

j m

where j m k and idx o dx In Thomassen gave the following

m j

Figure An Extended Caterpillar of Size Six

characterization of the strong semicomplete noneven digraphs

Let D b e a strong semicomplete digraph Then D is noneven if and only if D is a

sub digraph of an extended caterpillar

This characterization while valid if D has connectivity one neglects the connected digraph

which we shall call W due to its relationship with a certain wheel graph shown in Fig

ure W is strong semicomplete and noneven yet cannot b e a sub digraph of an extended

Figure An Extended Caterpillar of Size Six

caterpillar due to its connectedness We now state and prove a correct characterization for

the connected case In this pro of as well as in the Pro of of Theorem that follows we

have used ideas from the corresp onding pro ofs in

Theorem Let D b e a connected semicomplete noneven digraph Then D W

Pro of If n then D W since a connected noneven digraph of size four must have

exactly eight arcs with each vertex of indegree and outdegree two and W is the only such

semicomplete digraph

The pro of pro ceeds by induction Supp ose D has size n We rst consider the case

where D contains a vertex x for which D x is connected By the induction hypothesis

n and D x W in this case Since D is connected and semicomplete D must

contain as a sub digraph one of the digraphs shown in Figure Both of these digraphs

contain a weak doublecycle and thus in either case D is even which is a contradiction

Now supp ose that for every vertex x in D D x has connectivity one and is thus

a sub digraph of an extended caterpillar Let fu g b e the set of vertices from D x with

Figure See Pro of of Theorem

outdegree one and fv g the set of vertices from D x with indegree one Since any

sub digraph of an extended caterpillar has at least one vertex of each of these types b oth of

these sets are nonempty and since D x has size four or larger semicompleteness implies

that these two sets do not intersect Since D is connected x dominates each vertex in fv g

and is dominated by each vertex in fu g Given a vertex u fu g construct the digraph D

i i

by adding to D x all arcs from u to vertices in fv g Given a vertex v fv g construct the

i i

0 0

by adding to D x all arcs from vertices in fu g to v As noted in such a D digraph D

u v

or D is still noneven since the arc u v resp ectively u v can b e weighted identically

i i

to the path uxv resp ectively u xv

i i

0 0

Supp ose that for all u fu g and all v fv g D and D have connectivity one and x

i i

u v

u fu g so that D is a sub digraph of an extended caterpillar Since u has outdegree one

in D x u dominates at most one vertex from fv g If u dominates no vertex from fv g D

i i

has no vertex of indegree one a contradiction Thus u dominates exactly one vertex from

fv g Given any two vertices from fu g one of them must dominate the other since D x is

i i

semicomplete But this is a contradiction since each vertex of outdegree one must dominate

a vertex of indegree one This shows that D x contains exactly one vertex u of outdegree

one A similar argument shows that D x contains exactly one vertex v of indegree one

Since u dominates v and D x is a sub digraph of an extended caterpillar we conclude that

D x contains only two vertices which is a contradiction

is connected which implies that n Now supp ose there is a u fu g such that D

0 0

and D W Then u cannot dominate any v fv g in D x otherwise D would contain a

4 i

u u

vertex of indegree one contradicting its connectedness Thus by semicompleteness each

v fv g dominates u If there were more than one vertex of indegree one in D x then D

would contain two cycles incident at the same vertex which W do es not Thus there is

a unique v fv g and so D x is obtained from W by removing an arc from one of the

i 4

double cycles A similar argument yields the same form for D x in the case where there is

is connected Now since x dominates v and is dominated by u D a v fv g such that D

must b e isomorphic to one of the digraphs app earing in Figure Since b oth contain a weak

doublecycle we again obtain the contradiction that D is even completing the pro of

Figure See Pro of of Theorem

The second type of dense digraph is one in which the total degree of each vertex equals

or exceeds the size of D The characterization of the strong noneven digraphs in this class

from is as follows

If D is a strong noneven digraph with n vertices and of minimum degree at least n

then D is a sub digraph of an extended caterpillar or else D is isomorphic to C

This result is correct if D has connectivity one but neglects W a connected noneven

digraph which satises dv n for all its vertices A correct version for the connected

case follows

Theorem Supp ose D is a connected noneven digraph of size n such that dv n

for all its vertices v Then either D W or D C

as these are the only two noneven connected Pro of If n either D W or D C

digraphs of size four

Pro ceeding by induction supp ose D has size n By a result of Thomassen and

Haagkvist D must contain a cycle x x x which misses exactly one of its vertices

1 2 n1

Call the vertex not in this cycle x Since D is noneven it follows that there is at most one

cycle incident at x say from x Since x has degree n there must b e exactly one cycle

incident at x and every vertex of D x either dominates or is dominated by x If there exist

i j such that i j n and the arcs x x and x x are present then D contains

i j

a sub division of the digraph shown in Figure and is thus even which is a contradiction

Thus there is a k k n such that the vertices x x x dominate x

1 2 k

and the vertices x x x are dominated by x

k +1 k +2 n2

Construct the digraph D from D x as follows If each vertex of D x has degree at least

n then let D D x If D x has a vertex of degree n there must either b e a vertex

x i k such that x do es not dominate x or a vertex x k j n such

i i n1 j

that x do es not dominate x Form D by adding to D x either the arc x x or the

n1 j i n1

arc x x In either case D is noneven since the arc x x resp ectively x x

n1 j i n1 n1 j

Figure See Pro of of Theorem

can b e weighted identically to the path x xx resp ectively x xx in D Furthermore

i n1 n1 j

each vertex in D has degree at least n

We rst consider the case where D is connected By the induction hypothesis n

0 0

and D is either W or C If D D x C then D must contain the digraph shown in

4 4

Figure No matter how the undirected edges are oriented so long as D s connectedness

is required it is easily conrmed that D must contain a weak doublecycle contradicting

the fact that D is noneven If D D x W then D must contain one of the digraphs

shown in Figure and D is again even If D was obtained from D x by the addition

of an arc then D x must have had two vertices of degree three which is a contradiction

Thus D has connectivity one and is therefore a sub digraph of an extended caterpillar Let

the basic path of D b e z z z so that o d z id z

1 2 m 1 m

Claim The only vertex of outdegree one in D is z and the only vertex of indegree one in

Figure See Pro of of Theorem

D is z

Pro of of Claim D contains the hamiltonian cycle x x x x Since o dz and

1 2 n1 1 1

0 0

dz n every vertex in D other than z dominates z Thus the only vertex in D

1 1 1

other than z that could p ossibly have outdegree one is the vertex z which precedes z in the

1 0 1

hamiltonian cycle But if o dz then z dominates z so that z z This is imp ossible

0 1 0 0 2

A similar argument shows since n Thus z is the only vertex of outdegree one in D

that z is the only vertex of indegree one in D

Claim Supp ose z x Then the only vertices of indegree one in D x are z and

m n1 m

p ossibly x

0 0

Pro of of Claim If D D x or D is obtained from D x by the addition of the

arc x x j k then there is nothing to prove So supp ose D is obtained from

j n1

D x by adding the arc x x k j n Note that x z since otherwise

n1 j j m

idz in D x a contradiction since D x is strong If idx in D x then x

m j j

must dominate every other vertex in D x since dx n Thus x dominates z so

j j m

that x z Since z x dz n and z dominates z Noting that z

j m1 m n1 m m m1 m2

dominates z as well we see that x cannot have indegree one

m1 j

We can assume without loss of generality that z x since if this were the case we

1 n1

could consider the digraph obtained from D x by reversing every arc a digraph which is

noneven if and only if D x is

Claim In D x z do es not dominate x

1 n1

Pro of If z dominates x then x z Furthermore since the entire backbone

1 n1 n1 2

of D x must lie on the hamiltonian cycle x x x x z x Thus since z has

1 2 n1 1 1 n2 1

outdegree one z dominates x in D and so all vertices x i n dominate x in D as

1 i

well Thus in D o dx a contradiction since D is connected

Now form the digraph D from D x by adding all arcs of the form z y where y is

dominated by x in D Since z do es not dominate x in D x and x dominates x

1 n1 n1

00 00

in D each vertex v of D satises dv n Also D is noneven since the arc z y

00 00

in D can b e weighted the same as the path z xy in D Thus D is a sub digraph of an

extended caterpillar Now in D x z do esnt dominate z since n Furthermore x

1 m

dominates b oth x and x in D Thus if z x D has no vertex of indegree one

n1 m m n1

which is a contradiction If z x then either z is the only vertex of indegree one

m n1 m

in D x in which case the preceding argument leads to a contradiction or z is the only

vertex of outdegree one in D x Since D is obtained from D x by adding at least one

arc incident from z the latter case implies that D has no vertex of outdegree one which

is a contradiction that completes the pro of

Dense Noneven Digraphs and Symplectic Pairs

In order to simplify our characterization of the dense noneven digraphs allowing symplectic

pairs we rst explore a second necessary condition for a strong digraph to allow symplectic

pairs

Note that each nonzero term in the expansion of K i j corresp onds to a signed path

from j to i in D A see App endix A Recalling the denition of maximality of a sign

nonsingular pattern we shall refer to a noneven digraph as maximal if the addition of any

arc results in an even digraph

If a sign pattern H is signnonsingular we can use Lemma to construct the following

necessary condition for H to allow symplectic pairs

Prop osition Let the sign pattern H b e signnonsingular Then H can allow symplectic

pairs only if the following two conditions hold for all i j

H H i j is not signnonsingular

H H i j is signnonsingular

Pro of The prop osition follows directly from Lemma and the fact that a nonzero

entry in a signnonsingular pattern cannot corresp ond to a minor matrix whose determinant

changes sign

We now show that for a signnonsingular pattern H that allows symplectic pairs the

prop erties of maximality and irreducibility are equivalent In particular this implies that a

strong noneven digraph D that allows symplectic pairs must b e maximal

Prop osition Supp ose H is a negativediagonal signnonsingular pattern that allows

symplectic pairs Then H is maximal if and only if H is irreducible

Pro of First supp ose H is maximal Then by the denition of maximality H H i j

is not combinatorially singular Thus given any pair of distinct indices i j such that

H there is at least one simple path from j to i in D H Also by Prop osition

H H i j is signnonsingular again guaranteeing at least one simple path from j to

i in D H Thus D H is strongly connected and H is irreducible

Now supp ose H is irreducible By Prop osition H H i j is not sign

nonsingular Since H is irreducible D H is strongly connected so that H contains no

combinatorially singular minor matrix It follows that if H H i j s determinant

changes sign so that all zeros in H are necessary zeros H is therefore maximal

We now characterize the strong dense noneven digraphs that allow symplectic pairs

Theorem Supp ose D is a strong dense noneven digraph Then D allows symplectic

pairs if and only if D W D C or D is an extended caterpillar

are signequivalent Pro of As discussed in W allows symplectic pairs Since W and C

4 4

allows sym and since the symplectic pair prop erty is preserved under signequivalence C

then by Theorems and D is a plectic pairs as well If D is neither W nor C

sub digraph of an extended caterpillar First supp ose D is an extended caterpillar Then

since all extended caterpillars share the same indegree and outdegree sequences b oth of

which are f n n g D is signequivalent to the sp ecial caterpillar C all of

whose vertices are backbone vertices As shown in C allows symplectic pairs and so

D allows symplectic pairs as well Now supp ose D is a prop er sub digraph of an extended

caterpillar In this case D is not maximal since D was obtained by removing arcs from a

noneven digraph Thus D cannot allow symplectic pairs by Prop osition

Summary and Conclusions

Koh showed that there are innitely many noneven digraphs of minimum indegree and

outdegree two Thomassen showed that there are innitely many connected noneven

digraphs and Lim constructed an innite family of signnonsingular patterns asso ciated

with wheels whose digraphs are connected noneven and maximal planar Thomassen also

showed that no connected digraph is noneven

We have shown in this pap er that among the dense connected digraphs the only non

which are signequivalent As shown in even digraphs are of size four namely W and C

dense digraphs of connectivity one must b e sub digraphs of extended caterpillars The

only dense noneven digraphs that allow symplectic pairs are W C and extended cater

pillars For detailed characterizations of the symplectic pairs allowed by the sign patterns

corresp onding to these digraphs see and More generally no strong nonmaximal

noneven digraph can allow symplectic pairs

Among the connected symmetric digraphs the only noneven digraphs cite are the

digraphs G where G is a sub digraph of a C co ckade The only such digraph that allows

itself The only symmetric digraph of connectivity one that allows symplectic pairs is C

symplectic pairs is C

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Notes

CC Lim Binary Trees Symplectic Matrices and the Jacobi Co ordinates of Celestial

Mechanics Arch Rational Mech Anal

KM Koh Even Circuits in Directed Graphs and Lovasz Conjecture Bul letin of the

Malaysian Mathematical Society

C Thomassen The Even Cycle Problem for Directed Graphs Journal of the American

Mathematical Society Vol

D Schmidt Noneven Digraphs Symplectic Pairs and Full SignInvertibility PhD

Thesis