Log-concavity of the genus polynomials of Ringel ladders

0.48 MB·English

by  J. L. Gross

LOG-CONCAVITY OF THE GENUS POLYNOMIALS OF RINGEL
LADDERS
JONATHANL.GROSS,TOUFIKMANSOUR,THOMASW.TUCKER,
5
ANDDAVIDG.L.WANG
1
0
2 Abstract. A Ringel ladder can be formed by a self-bar-amalgamation op-

n eration on a symmetric ladder, that is, by joining the root vertices on its
a end-rungs. The present authors have previously derived criteria under which
J linear chains ofcopiesofoneormoregraphshavelog-concavegenuspolyno-
5 mials. Herein we establish Ringel ladders as the first significant non-linear
2 infinite family of graphs known to have log-concave genus polynomials. We
construct an algebraic representation of self-bar-amalgamation as a matrix
] operation, to be applied to a vector representation of the partitioned genus
O distribution ofasymmetricladder. Analysisoftheresultinggenuspolynomial
involves the use of Chebyshev polynomials. This paper continues our quest
C
to affirm the quarter-century-old conjecture that all graphs have log-concave
h. genuspolynomials.
t
a
m
[
1. Genus Polynomials
1
v
Our graphs are implicitly taken to be connected, and our graph embeddings
6
are cellular and orientable. For general background in topological graph theory,
0
1 see [13, 1]. Prior acquaintance with the concepts of partitioned genus distribution
6 (abbreviated here as pgd) and production (e.g., [10, 17]) are necessary preparation
0 for reading this paper. The exposition here is otherwise intended to be accessible
.
1 both to graph theorists and to combinatorialists.
0
The number of combinatorially distinct embeddings of a graph G in the ori-
5
entable surface of genus i is denoted by g (G). The sequence g (G), g (G), g (G),
1 i 0 1 2
: ..., is called the genus distribution of G. A genus distribution contains only
v
finitely many positive numbers, and there are no zeros between the first and last
i
X positive numbers. The genus polynomial is the polynomial
ar ΓG(x) = g0(G)+g1(G)x+g2(G)x2+... .
Log-concave sequences
A sequence A = (a )n is said to be nonnegative, if a ≥ 0 for all k. An
k k=0 k
element a is said to be an internal zero of A if a =0 and if there exist indices
k k
i and j with i < k < j, such that a a (cid:54)= 0. If a a ≤ a2 for all k, then A is
i j k−1 k+1 k
said to be log-concave. If there exists an index h with 0≤h≤n such that
a ≤ a ≤ ··· ≤ a ≤ a ≥ a ≥ ··· ≥ a ,
0 1 h−1 h h+1 n
2000 Mathematics Subject Classification. 05A15,05A20,05C10.
1
2 J.L.GROSS,T.MANSOUR,T.W.TUCKER,ANDD.G.L.WANG
then A is said to be unimodal. It is well-known that any nonnegative log-concave
sequence without internal zeros is unimodal, and that any nonnegative unimodal
sequence has no internal zeros. A prior paper [11] by the present authors provides
additional contextual information regarding log-concavity and genus distributions.
For convenience, we sometimes abbreviate the phrase “log-concave genus distri-
bution”asLCGD.Proofsthatclosed-endladdersanddoubledpathshaveLCGDs
[4] were based on explicit formulas for their genus distributions. Proof that bou-
quetshaveLCGDs[12]wasbasedonarecursion. Aconjecturethatallgraphshave
LCGDs was published by [12].
Stahl’s method [21, 22] of representing what we have elsewhere formulated as
simultaneous recurrences [4] or as a transposition of a production system for a
surgical operation on graph embeddings as a matrix of polynomials can simplify a
proof that a family of graphs has log-concave genus distributions, without having
to derive the genus distribution itself.
Newton’s theorem that real-rooted polynomials with non-negative coefficients
are log-concave is one way of getting log-concavity. Stahl [22] made the general
conjecture(Conjecture6.4)thatallgenuspolynomialsarereal-rooted, andhegave
a collection of specific test families. Shortly thereafter, Wagner [24] proved that
the genus distributions for the related closed-end ladders and various other test
families suggested by [22] are real-rooted. However, Liu and Wang [16] answered
Stahl’s general conjecture in the negative, by exhibiting a chain of copies of the
wheel graph W , one of Stahl’s test families, that is not real-rooted. Our previous
4
paper [11] proves, nonetheless, that the genus distribution of every graph in the
W -linear sequence is log-concave. Thus, even though Stahl’s proposed approach
4
to log-concavity via roots of genus polynomials is sometimes infeasible, results in
[11] do support Stahl’s expectation that chains of copies of a graph are a relatively
accessible aspect of the general LCGD problem. The genus distributions for the
family of Ringel ladders, whose log-concavity is proved in this paper, are not real-
rooted either.
Log-concavity of genus distributions for directed graph embeddings has been
studied by [2] and [3]. Another related area is the continuing study of maximum
genus of graphs, of which [15] is an example.
Linear, ringlike, and tree-like families
Stahl used the term“H-linear” to describe chains of graphs that are constructed
by amalgamating copies of a fixed graph H. Such amalgamations are typically
on a pair of vertices, one in each of the amalgamands, or on a pair of edges. It
seems reasonable to generalize the usage of linear in several ways, for instance, by
allowing graphs in the chain to be selected from a finite set.
We use the term ring-like to describe a graph that results from any of the
following topological operations on a doubly rooted linear chain with one root in
the first graph of the chain and one in the last graph:
(1) a self-amalgamation of two root-vertices;
(2) a self-amalgamation of two root-edges;
(3) joiningoneroot-vertextotheotherroot-vertex(whichiscalledaself-bar-
amalgamation).
Every graph can be regarded as tree-like in the sense of tree decompositions.
However,weusethistermonlywhenagraphisnotlinearorring-like. Foranyfixed
LOG-CONCAVITY OF GENUS POLYNOMIALS OF RINGEL LADDERS 3
tree-width w and fixed maximum degree ∆, there is a quadratic-time algorithm [8]
to calculate the genus polynomial of graphs of parameters w and ∆. One plausible
approach to the general LCGD conjecture might be to prove it for fixed tree-
width and fixed maximum degree. Recurrences have been given for the the genus
distributionsofcubicouterplanargraphs[6],4-regularouterplanargraphs[18],and
cubic Halin graphs [7], all three of which are tree-like. However, none of these
genus distributions have been proved to be log-concave. Nor have any other tree-
like graphs been proved to have LCGDs.
This paper is organized as follows. Section 2 describes a representation of parti-
tioning of the genus distribution into ten parts as a pgd-vector. Section 3 describes
how productions are used to describe the effect of a graph operation on the pgd-
vector. Section4analyzeshowself-baramalgamationaffectsthegenusdistribution.
Section 5 offers a new derivation of the genus distributions of the Ringel ladders
and proof that these genus distributions are log-concave.
2. Partitioned Genus Distributions
Afundamentalstrategyinthecalculationofgenusdistributions,fromtheoutset
[4], has been to partition a genus distribution according to the incidence of face-
boundary walks on one or more roots. We abbreviate “face-boundary walk” as
fb-walk. Foragraph(G,u,s)withtwo2-valentroot-vertices, wecanpartitionthe
number g (G) into the following four parts:
i
dd (G): thenumberofembeddingsof(G,u,v)inthesurfaceS suchthattwo
i i
distinct fb-walks are incident on root u and two on root v;
ds (G): the number of embeddings in S such that two distinct fb-walks are
i i
incident on root u and only one on root v;
sd (G): thenumberofembeddingsinS suchthatonefb-walkistwiceincident
i i
on root u and two distinct fb-walks are incident on root v;
ss (G): thenumberofembeddingsinS suchthatonefb-walkistwiceincident
i i
on root u and one is twice incident on root v.
Clearly, g (G) = dd (G) + ds (G) + sd (G) + ss (G). Each of the four parts is
i i i i i
sub-partitioned:
dd0(G): thenumberoftype-ddembeddingsof(G,u,v)inS suchthatneither
i i
fb-walk incident at root u is incident at root v;
dd(cid:48)(G): the number of type-dd embeddings in S such that one fb-walk inci-
i i
dent at root u is incident at root v;
dd(cid:48)(cid:48)(G): the number of type-dd embeddings in S such that both fb-walks
i i
incident at root u are incident at root v;
ds0(G): the number of type-ds embeddings in S such that neither fb-walk
i i
incident at root u is incident at root v;
ds(cid:48)(G): thenumberoftype-dsembeddingsinS suchthatonefb-walkincident
i i
at root u is incident at root v;
sd0(G): thenumberoftype-sdembeddingsinS suchthatthefb-walkincident
i i
at root u is not incident on root v;
sd(cid:48)(G): thenumberoftype-sdembeddingsinS suchthatthefb-walkatroot
i i
u is also incident at root v;
4 J.L.GROSS,T.MANSOUR,T.W.TUCKER,ANDD.G.L.WANG
ss0(G): thenumberoftype-ssembeddingsinS suchthatthefb-walkincident
i i
at root u is not incident on root v;
ss1(G): thenumberoftype-ssembeddingsinS suchthatthefb-walkincident
i i
at root u is incident at root v, and the incident pattern is uuvv;
ss2(G): thenumberoftype-ssembeddingsinS suchthatthefb-walkincident
i i
at root u is incident at root v, and the incident pattern is uvuv.
We define the pgd-vector of the graph(G,u,v) to be the vector
(cid:0)dd(cid:48)(cid:48)(G) dd(cid:48)(G) dd0(G) ds0(G) ds(cid:48)(G)
sd0(G) sd(cid:48)(G) ss0(G) ss1(G) ss2(G)(cid:1)
with ten coordinates, each a polynomial in x. For instance,
ds(cid:48)(G) = ds˜(G) + ds(cid:48)(G)x + ds(cid:48)(G)x2 + ··· .
0 1 2
3. Symmetric Ladders
We define the symmetric ladder (L¨ ,u,v) to be the graph obtained from the
n
cartesian product P (cid:50)P by contracting the respective edges at both ends that
2 n+2
join a pair of 2-valent vertices and designating the remaining two 2-valent vertices
at the ends of the ladder as root-vertices. The symmetric ladders (L¨ ,u,v) and
1
(L¨ ,u,v) are illustrated in Figure 3.1. The location of the roots of a symmetric
2
ladder at opposite ends causes it to have a different partitioned genus distribution
from other ladders to which it is isomorphic when the roots are disregarded.
v u v
u
: :
L L
1 2
Figure 3.1. The symmetric ladders L¨ and L¨ .
1 2
Productions
A production is an algebraic representation of the set of possible effects of a
graphoperationonagraphembedding. Forinstance,addingarungtoanembedded
symmetricladder(L¨ ,u,v)involvesinsertinganewvertexoneachsideoftheroot-
n
vertexvandthenjoiningthetwonewvertices. Sinceboththeresultingnewvertices
are trivalent, the number of embeddings of (L¨ ,u,v) that can result is 4. Thus,
n+1
the sum of the coefficients in the consequent of the production (the right side) is
4. Figures 3.2 and 3.3 are topological derivations of the following ten productions
usedtoderivethepartitionedgenusdistributionof(L¨ ,u,v)fromthepartitioned
n+1
genus distribution of (L¨ ,u,v).
n
dd0 −→ 2dd0+2sd0
i i i+1
dd(cid:48) −→ dd0+dd(cid:48) +2sd(cid:48)
i i i i+1
dd(cid:48)(cid:48) −→ 2dd(cid:48) +2ss2
i i i+1
ds0 −→ 2ds0+2ss0
i i i+1
ds(cid:48) −→ ds0+ds(cid:48) +2ss1
i i i i+1
LOG-CONCAVITY OF GENUS POLYNOMIALS OF RINGEL LADDERS 5
sd0 −→ 4dd0
i i
sd(cid:48) −→ 4dd(cid:48)
i i
ss0 −→ 4ds0
i i
ss1 −→ 4ds(cid:48)
i i
ss2 −→ 2ds(cid:48) +2dd(cid:48)(cid:48)
i i i
v
dd0 -> 2dd0 + 2sd0
i i i+1
u
v
dd' -> dd' +dd0+ 2sd'
i i i i+1
u
v
dd" -> 2dd' + 2ss2
i i i+1
u
v
ds0 -> 2ds0 + 2ss0
i i i+1
u
v
ds' -> ds0 + ds' + 2ss1
i i i i+1
u
Figure 3.2. Five productions for construction of symmetric ladders.
6 J.L.GROSS,T.MANSOUR,T.W.TUCKER,ANDD.G.L.WANG
v
sd0 -> 4dd0
i i
u
sd' -> 4dd'
i i
v
ss0 -> 4ds0
i i
u
v
ss1 -> 4ds'
i i
u
v
ss2 -> 2ds' + 2dd"
i i i
u
Figure 3.3. Five more productions for symmetric ladders.
LOG-CONCAVITY OF GENUS POLYNOMIALS OF RINGEL LADDERS 7
Theorem 3.1. The pgd-vector of the symmetric ladder (L¨ ,u,v) is
0
(cid:0) (cid:1)
(3.1) V = 0 0 1 0 0 0 0 0 0 0 .
L0
For n > 0, the pgd-vector of the symmetric ladder (L¨ ,u,v) is the product of the
n
row-vector V with the 10×10 production matrix
Ln−1
 
2 0 0 0 0 2x 0 0 0 0
1 1 0 0 0 0 2x 0 0 0
 
0 2 0 0 0 0 0 0 0 2x
 
0 0 0 2 0 0 0 2x 0 0
 
0 0 0 1 1 0 0 0 2x 0
(3.2) M =  
4 0 0 0 0 0 0 0 0 0
 
0 4 0 0 0 0 0 0 0 0
 
0 0 0 4 0 0 0 0 0 0
 
0 0 0 0 4 0 0 0 0 0
0 0 2 0 2 0 0 0 0 0
Proof. EachofthetenrowsofthematrixM representsoneofthetenproductions.
For instance, the first two rows represent the productions
dd0 −→ 2dd0+2sd0
i i i+1
dd(cid:48) −→ dd0+dd(cid:48) +2sd(cid:48) (cid:3)
i i i i+1
Example 3.1. We iteratively calculate pgd-vectors of the symmetric ladders L
n
for n≤4
V = (0 0 1 0 0 0 0 0 0 0)
L0
V = (0 2 0 0 0 0 0 0 0 2x)
L1
V = (2 2 4x 0 4x 0 4x 0 0 0)
L2
V = (6 2+24x 0 4x 4x 4x 4x 0 8x2 8x2)
L3
V = (14+40x 2+40x 16x2 12x 4x+48x2 12x 4x+48x2 8x2 8x2 0)
L4
4. Self-Bar-Amalgamations
We recall from Section 1 that the self-bar-amalgamation of any doubly vertex-
rootedgraph(G,u,v), whichisdenoted ∗ (G,u,v), isformedbyjoiningtheroots
uv
u and v. The present case of interest is when the two roots are 2-valent and non-
adjacent. We observe that if G is a cubic 2-connected graph and if each of the two
roots is created by placing a new vertex in the interior of an edge of G, then the
result of the self-bar amalgamation is again a 2-connected cubic graph.
Theorem 4.1. Let(G,u,v)beagraphwithtwonon-adjacent2-valentvertexroots.
The (non-partitioned) genus distribution of the graph ∗ (G,u,v), obtained by self-
uv
bar-amalgamation, can be calculated as the dot-product of the pgd-vector V with
G
the following row-vector:
(cid:0) (cid:1)
(4.1) B = 4x 1+3x 2+2x 4x 2+2x 4x 2+2x 4x 4 4 .
8 J.L.GROSS,T.MANSOUR,T.W.TUCKER,ANDD.G.L.WANG
Proof. Figures 4.1 and 4.2 derive the ten corresponding productions.
dd0 −→4g
i i+1
dd(cid:48) −→g +3g
i i i+1
dd(cid:48)(cid:48) −→2g +2g
i i+1
ds0 −→4g
i i+1
ds(cid:48) −→2g +2g
i i i+1
sd0 −→4g
i i+1
sd(cid:48) −→2g +2g
i i i+1
ss0 −→4g
i i+1
ss1 −→4g
i i
ss2 −→4g (cid:3)
i i
v
dd0 -> 4g
i i+1
u
v
dd' -> g + 3g
i i i+1
u
v
dd'' -> 2g + 2g
i i i+1
u
v
ds0 -> 4g
i i+1
u
v
ds' -> 2g + 2g
i i i+1
u
Figure 4.1. Five productions for self-bar-amalgamation.
LOG-CONCAVITY OF GENUS POLYNOMIALS OF RINGEL LADDERS 9
sd0 -> 4g
i i+1
sd' -> 2g + 2g
i i i+1
ss0 -> 4g
i i+1
ss1 -> 4g
i i
ss2 -> 4g
i i
Figure 4.2. Five more productions for self-bar-amalgamation.
5. Ringel Ladders
We define a Ringel ladder RL to be the result of a self-bar-amalgamation on
n
the symmetric ladder (L¨ ,u,v). Such ladders were introduced by Gustin [14] and
n
usedextensivelybyRingel[19]inhissolutionwithYoungs[20]oftheHeawoodmap-
coloring problem. The Ringel ladder RL is illustrated in Figure 5.1. The genus
4
distributions of Ringel ladders were first calculated by Tesar [23]. Our rederivation
here is to facilitate our proof of their log-concavity.
Figure 5.1. The Ringel ladder RL .
4
10 J.L.GROSS,T.MANSOUR,T.W.TUCKER,ANDD.G.L.WANG
Example 5.1. We take dot products of the pgd-vectors calculated in Example 3.1
V = (0 0 1 0 0 0 0 0 0 0)
L0
V = (0 2 0 0 0 0 0 0 0 2x)
L1
V = (2 2 4x 0 4x 0 4x 0 0 0)
L2
V = (6 2+24x 0 4x 4x 4x 4x 0 8x2 8x2)
L3
V = (14+40x 2+40x 16x2 12x 4x+48x2 12x 4x+48x2 8x2 8x2 0)
L4
with the vector (4.1)
(cid:0) (cid:1)
B = 4x 1+3x 2+2x 4x 2+2x 4x 2+2x 4x 4 4
to obtain the genus polynomials of the corresponding Ringel ladders.
Γ (x) = 2+2x
RL0
Γ (x) = 2+14x
RL1
Γ (x) = 2+38x+24x2
RL2
Γ (x) = 2+70x+184x2
RL3
Γ (x) = 2+118x+648x2+256x3
RL4
Theorem 5.1. The genus distribution of the Ringel ladder RL is given by taking
n
the dot product of the vector B with the product of the vector V and the matrix
L0
Mn, where B is given by (4.1), and M is given by (3.2).
Proof. This follows immediately from Theorem 4.1. (cid:3)
To deduce an explicit expression of Γ (x), we shall use Chebyshev polynomi-
RLn
als. Chebyshev polynomials of the second kind are defined by the recurrence
relation
U (x) = 2xU (x)−U (x),
p p−1 p−2
with U (x) = 1 and U (x) = 2x. It can be equivalently defined by the generating
0 1
function
(cid:88) 1
(5.1) U (x)tp = .
p 1−2xt+t2
p≥0
The pth Chebyshev polynomial U (x) can be expressed by
p
(cid:18) (cid:19)
(cid:88) p−j
U (x)= (−1)j (2x)p−2j.
p j
j≥0
Theorem 5.2. The genus distribution of the Ringel ladder RL is given by
n
(cid:18)(cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
(cid:88) n−j n−j+1
Γ (x) = (1−x) + (8x)j
RLn j j
j≥0
(cid:18)(cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
(cid:88) n−j n−j+1
+x2n+1 + (2x)j.
j j
j≥0
Proof. Using Theorem 5.1 and mathematical software such as Maple, we calculate
the generating function
(cid:88)
V Mntn =(a,b,c,2xta,2xtb,2xta,2xtb,4x2t2a,4x2t2b,2xtc),
L0
n≥0