A construction of small (q-1)-regular graphs of girth 8

0.33 MB·English

by  M. Abreu

A construction of small (q 1)-regular graphs of girth 8
−
M. Abreu1, G. Araujo-Pardo2, C. Balbuena3, D. Labbate1 ∗
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1 1Dipartimento di Matematica, Informatica ed Economia, Universita` degli Studi della Basilicata,
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2 Viale dell’Ateneo Lucano, I-85100 Potenza, Italy.
2
n Instituto de Matema´ticas, Universidad Nacional Auto´noma de M´exico,
a
J M´exico D. F., M´exico
1 3Departament de Matema´tica Aplicada III, Universitat Polit`ecnica de Catalunya,
1
Campus Nord, Edifici C2, C/ Jordi Girona 1 i 3 E-08034 Barcelona, Spain.

]
O
C
.
h
Abstract
t
a
m
In this note we construct a new infinite family of (q 1)-regular graphs of girth 8 and
−
[ order 2q(q 1)2 for all prime powers q 16, which are the smallest known so far whenever

− ≥
1 q 1 is not a prime power or a prime power plus one itself.
v −
2
5 Keywords: Cages, girth, Moore graphs, perfect dominating sets.
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2 MSC2010: 05C35, 05C69
0
.
1
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5 1 Introduction
1
:
v
i Throughout this note, only undirected simple graphs without loops or multiple edges are con-
X
sidered. Unless otherwise stated, we follow the book by Bondy and Murty [11] for terminology
r
a
and notation.
Let G be a graph with vertex set V = V(G) and edge set E = E(G). The girth of a graph
G is the number g = g(G) of edges in a smallest cycle. For every v V, N (v) denotes the
G
∈
neighbourhood of v, that is, the set of all vertices adjacent to v. The degree of a vertex v V
∈
is the cardinality of N (v). A graph is called regular if all the vertices have the same degree.
G
A (k,g)-graph is a k-regular graph with girth g. Erd˝os and Sachs [12] proved the existence of
∗Research supported by the Ministerio de Educacio´n y Ciencia, Spain, the European Regional Development
Fund (ERDF) under project MTM2011-28800-C02-02; by the Catalonian Government under project 1298
SGR2009;byCONACyT-M´exicounderproject57371;byPAPIIT-M´exicounderproject104609-3;bytheItalian
Ministry MIUR and carried out within the activity of INdAM-GNSAGA.
Email addresses: [email protected] (M. Abreu), [email protected] (G. Araujo),
[email protected] (C. Balbuena), [email protected] (D. Labbate)
1
(k,g)-graphs for all values of k and g provided that k 2. Since then most work carried out
≥
has focused on constructing a smallest one (cf. e.g. [1, 2, 3, 4, 5, 6, 7, 9, 13, 15, 18, 20, 21]).
A (k,g)-cage is a k-regular graph with girth g having the smallest possible number of vertices.
CageshavebeenintenselystudiedsincetheywereintroducedbyTutte[23]in1947. Moredetails
about constructions of cages can be found in the recent survey by Exoo and Jajcay [14].
Inthis note weare interestedin(k,8)-cages. Countingthenumberofvertices inthedistance
partition with respect to an edge yields the following lower bound on the order of a (k,8)-cage:
n (k,8) = 2(1+(k 1)+(k 1)2+(k 1)3). (1)
0
− − −
A (k,8)-cage with n (k,8) vertices is called a Moore (k,8)-graph (cf. [11]). These graphs
0
have been constructed as the incidence graphs of generalized quadrangles of order k 1 (cf. [9]).
−
All these objects are known to exist for all prime power values of k 1 (cf. e.g. [8, 16]), and
−
no example is known when k 1 is not a prime power. Since they are incidence graphs, these
−
cages are bipartite and have diameter 4.
AsubsetU V(G)issaidtobea perfect dominating set of Gifforeachvertexx V(G) U,
⊂ ∈ \
N (x) U = 1(cf. [17]). NotethatifGisa(k,8)-graphandU isaperfectdominatingsetofG,
G
| ∩ |
thenG U isclearlya(k 1,8)-graph. Usingclassicalgeneralizedquadrangles, Beukemannand
− −
Metsch[10]provedthatthecardinalityofaperfectdominatingsetB ofaMoore(q+1,8)-graph,
q a prime power, is at most B 2(2q2+2q) and if q is even B 2(2q2+q+1).
| | ≤ | | ≤
For k = q+1 where q 2 is a prime power, we find a perfect dominating set of cardinality
≥
2(q2+3q+1) for all q (cf. Proposition 2.1). This result allows us to explicitly obtain q-regular
graphs of girth 8 and order 2q(q2 2) for any prime power q (cf. Definition 2.2). Finally,
−
we prove the existence of a perfect dominating set of these q-regular graphs which allow us to
construct a new infinite family of (q 1)-regular graphs of girth 8 and order 2q(q 1)2 for all
− −
prime powers q (cf. Theorem 2.1), which are the smallest known so far for q 16 whenever
≥
q 1 is not a prime power or a prime power plus one itself. Previously, the smallest known
−
(q 1,8)-graphs, for q a prime power, were those of order 2q(q2 q 1) which appeared in [7].
− − −
The first ten improved values appear in the following table in which k = q 1 is the regularity
−
of a (k,8)–graph, and the other columns contain the old and the new upper bound on its order.
k Bound in [7] New bound k Bound in [7] New bound
15 7648 7200 52 292030 286624
22 23230 22264 58 403678 396952
36 98494 95904 63 515968 508032
40 134398 131200 66 592414 583704
46 203134 198904 70 705598 695800
2
2 Construction of small (q 1)-regular graphs of girth 8
−
In this section we construct (q 1)-regular graphs of girth 8 with 2q(q 1)2 vertices, for every
− −
prime power q 4. To this purpose we need the following coordinatization of a Moore (q+1,8)-
≥
cage Γ .
q
Definition 2.1 [19, 22] Let F be a finite field with q 2 a prime power and (cid:37) a sym-
q
≥
bol not belonging to F . Let Γ = Γ [V ,V ] be a bipartite graph with vertex sets V =
q q q 0 1 i
F3 ((cid:37),b,c) ,((cid:37),(cid:37),c) : b,c F ((cid:37),(cid:37),(cid:37)) , i = 0,1, and edge set defined as follows:
q ∪{ i i ∈ q}∪{ i}
For all a F (cid:37) and for all b,c F :
q q
∈ ∪{ } ∈

(w, aw+b, a2w+2ab+c) : w F ((cid:37),a,c) if a F ;
 0 q 0 q
{ ∈ }∪{ } ∈
N ((a,b,c) ) =
Γq 1
 (c,b,w) : w F ((cid:37),(cid:37),c) if a = (cid:37).
0 q 0
{ ∈ }∪{ }
N (((cid:37),(cid:37),c) ) = ((cid:37),c,w) : w F ((cid:37),(cid:37),(cid:37))
Γq 1 { 0 ∈ q}∪{ 0}
N (((cid:37),(cid:37),(cid:37)) ) = ((cid:37),(cid:37),w) : w F ((cid:37),(cid:37),(cid:37)) .
Γq 1 { 0 ∈ q}∪{ 0}
Or equivalently
For all i F (cid:37) and for all j,k F :
q q
∈ ∪{ } ∈

(w, j wi, w2i 2wj +k) : w F ((cid:37),j,i) if i F ;
 1 q 1 q
{ − − ∈ }∪{ } ∈
N ((i,j,k) ) =
Γq 0
 (j,w,k) : w F ((cid:37),(cid:37),j) if i = (cid:37).
1 q 1
{ ∈ }∪{ }
N (((cid:37),(cid:37),k) ) = ((cid:37),w,k) : w F ((cid:37),(cid:37),(cid:37)) ;
Γq 0 { 1 ∈ q}∪{ 1}
N (((cid:37),(cid:37),(cid:37)) ) = ((cid:37),(cid:37),w) : w F ((cid:37),(cid:37),(cid:37)) .
Γq 0 { 1 ∈ q}∪{ 1}
Notethat(cid:37)isjustasymbolnotbelongingtoF andnoarithmeticaloperationwillbeperformed
q
with it. Figure 1 shows a spanning tree of Γ with the vertices labelled according to Definition
q
2.1.
Proposition 2.1 Let q 2 be a prime power and let Γ = Γ [V ,V ] be the Moore (q +1,8)-
q q 0 1
≥
graph with the coordinatization in Definition 2.1. Let A = ((cid:37),0,c) : c F ((cid:37),(cid:37),0) and
1 q 1
{ ∈ }∪{ }
let x F 0 . Then the set
q
∈ \{ }
(cid:32) (cid:33)
(cid:92)
N [A] N2 (a) N2 [((cid:37),(cid:37),x) ]
Γq ∪ Γq ∪ Γq 1
a∈A
is a perfect dominating set of Γ of cardinality 2(q2+3q+1).
q
3
(̺,̺,̺)1 (̺,̺,̺)0
(̺,̺,0)0 ······ (̺,̺,i)0 (̺,̺,0)1 ······ (̺,̺,a)1
(̺,0,0)1 (̺,j,0)1 (̺,0,i)1 (̺,j,i)1 (̺,0,0)0 (̺,0,c)0 (̺,a,0)0 (̺,a,c)0
··· ··· ··· ···
··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ···
(0,0,0)0(0,0,w)0(0,j,0)0(0,j,w)0(i,0,0)0(i,0,w)0(i,j,0)0(i,j,w)0(0,0,0)1(0,t,0)1 (0,0,c)1(0,t,c)1 (a,0,0)1(a,t,0)1(a,0,c)1(a,t,c)1
Figure 1: Spanning tree of Γ .
q
Proof From Definition 2.1, it follows that A = ((cid:37),0,c) : c F ((cid:37),(cid:37),0) has cardinality
1 q 1
{ ∈ }∪{ }
q+1anditselementsaremutuallyatdistancefour. Then N [A] = (q+1)2+q+1. ByDefinition
| Γq |
2.1, N (((cid:37),0,c) ) = (c,0,w) : w F ((cid:37),(cid:37),c) ; and N (((cid:37),(cid:37),0) ) = ((cid:37),0,w) : w
Γq 1 { 0 ∈ q}∪{ 0} Γq 1 { 0 ∈
F ((cid:37),(cid:37),(cid:37)) . Then ((cid:37),(cid:37),(cid:37)) N2 (((cid:37),0,c) )) N2 (((cid:37),(cid:37),0) )) for all c F . Moreover,
q}∪{ 0} 1 ∈ Γq 1 ∩ Γq 1 ∈ q
N ((c,0,w) ) = (a, ac,a2c+w) : a F ((cid:37),0,c) . Thus, for all c ,c ,w ,w F ,
Γq 0 { − 1 ∈ q}∪{ 1} 1 2 1 2 ∈ q
c = c , we have (a, c a,a2c +w ) = (a, c a,a2c +w ) if and only if a = 0 and w = w .
1 2 1 1 1 1 2 2 2 1 1 2
(cid:54) − −
Let I = (cid:84) N2 (a). We conclude that I = ((cid:37),(cid:37),(cid:37)) (0,0,w) : w F which implies
A a∈A Γq A { 1}∪{ 1 ∈ q}
that N [A] + I = (q+1)2+2(q+1).
| Γq | | A|
Since N2 [((cid:37),(cid:37),x) ] = (cid:83) N [((cid:37),x,j) ] N [((cid:37),(cid:37),(cid:37)) ] we obtain that (N [A] I )
Γq 1 j∈Fq Γq 0 ∪ Γq 0 Γq ∪ A ∩
N2 [((cid:37),(cid:37),x) ] = ((cid:37),(cid:37),(cid:37)) ,((cid:37),(cid:37),0) ,((cid:37),(cid:37),(cid:37)) . Let D = N [A] I N2 [((cid:37),(cid:37),x) ], then
Γq 1 { 0 1 1} Γq ∪ A∪ Γq 1
D = N [A] + I + N2 [((cid:37),(cid:37),x) ] 3
| | | Γq | | A| | Γq 1 |−
= (q+1)2+2(q+1)+1+(q+1)+q(q+1) 3
−
= 2q2+6q+2.
Let us prove that D is a perfect dominating set of Γ .
q
Let H denote the subgraph of Γ induced by D. Note that for t,c F , the vertices
q q
∈
(x,t,c) N2 (((cid:37),(cid:37),x) ) have degree 2 in H because they are adjacent to the vertex ((cid:37),x,t)
1 ∈ Γq 1 0 ∈
N ((cid:37),(cid:37),x) and also to the vertex ( x−1t,0,xt+z) N (A). This implies that the vertices
Γq 1 − 0 ∈ Γq
(i,0,j) N (A), i,j F , havedegree3inH and, alsothatthediameterofH is5. Moreover,
0 ∈ Γq ∈ q
fork F , thevertices((cid:37),(cid:37),k) ,((cid:37),0,k) D havedegree2inH andthevertices((cid:37),(cid:37),j) D,
q 0 0 1
∈ ∈ ∈
j F 0,x have degree 1 in H. All other vertices in D have degree q+1 in H.
q
∈ \{ }
Since the diameter of H is 5 and the girth is 8, N (v) D 1 for all v V(Γ ) D,
| Γq ∩ | ≤ ∈ q \
and also for all distinct d,d(cid:48) D we have (N (d) N (d(cid:48))) (V(Γ ) D) = . Then,
∈ Γq ∩ Γq ∩ q \ ∅
N (D) (V(Γ ) D) = q2(q 2)+2q(q 1)+(q 2)q+q2(q 1) = 2q3 4q = V(Γ ) D .
| Γq ∩ q \ | − − − − − | q \ |
Hence N (v) D = 1 for all v V(Γ ) D. Thus D is a perfect dominating set of Γ .
| Γq ∩ | ∈ q \ q
4
Definition 2.2 Let q 4 be a prime power and let x F 0,1 . Define Gx as the q-regular
≥ ∈ q \{ } q
graph of girth 8 and order 2q(q2 2) constructed in Proposition 2.1.
−
Theorem 2.1 Let q 4 be a prime power and let Gx be the graph given in Definition 2.2. Let
≥ q
R = N ( ((cid:37),j,k) : j,k F ,j = 0,1,x ) N5 (((cid:37),1,0) ). Then, the set
Gxq { 0 ∈ q (cid:54) } ∩ Gxq 0
(cid:91)
S := N [((cid:37),1,j) ] N [R]
Gx 0 Gx
q ∪ q
j∈Fq
is a perfect dominating set in Gx of cardinality 4q2 6q. Hence, Gx S is a (q 1)-regular
q − q − −
graph of girth 8 and order 2q(q 1)2.
−
Proof Once x F 0,1 has been chosen to define Gx, to simplify notation, we will denote
∈ q \{ } q
Gx by G throughout the proof. Denote by P = ((cid:37),j,k) : j,k F ,j = 0,1,x , then R =
q q { 0 ∈ q (cid:54) }
N (P) N5 (((cid:37),1,0) ). Notethatd (((cid:37),1,0) ,((cid:37),j,k) ) = 4, becauseaccordingtoDefinition
Gq ∩ Gq 0 Gq 0 0
2.1, G containsthefollowingpathsoflengthfour(seeFigure2): ((cid:37),1,0) (1,b,0) (w,w+b,w+
q 0 1
2b) (j,t,k) ((cid:37),j,k) , for all b,j,t F such that b+w = 0 due to the vertices (j,0,k) with
0 1 0 q 0
∈ (cid:54)
second coordinate zero have been removed from Γ to obtain G . By Definition 2.1 we have
q q
w+b = jw+t and w+2b = j2w+2jt+k. If w+b = 0, then w = b = tj−1 and b = jt+k
−
yielding that t = (1 j2)−1jk. This implies that (j,(1 j2)−1jk,k) R is the unique neighbor
1
− − ∈
in R of ((cid:37),j,k) P. Therefore every ((cid:37),j,k) P has a unique neighbor (j,t,k) R leading
0 0 1
∈ ∈ ∈
to:
R = P = q(q 3). (2)
| | | | −
Thus, every v N (R) P has at most R /q = q 3 neighbors in R because for each j
∈ Gq \ | | −
the vertices from the set ((cid:37),j,k) : k F P are mutually at distance 6 (they were the q
0 q
{ ∈ } ⊂
neighborsinΓ oftheremovedvertex((cid:37),(cid:37),j) ). Furthermore, everyv N (R) P hasatmost
q 1 ∈ Gq \
one neighbor in N5 (((cid:37),1,0) ) R because the vertices ((cid:37),1,j) : j F ,j = 0 are mutually
Gq 0 \ { 0 ∈ q (cid:54) }
at distance 6. Therefore every v N (R) P has at least two neighbors in N3 (((cid:37),1,0) ).
∈ Gq \ Gq 0
Thus denoting K = N (N (R) P) N3 (((cid:37),1,0) ) we have
Gq Gq \ ∩ Gq 0
K 2 N (R) P . (3)
| | ≥ | Gq \ |
Moreover, observe that (N (P) R) K = because these two sets are at distance four
Gq \ ∩ ∅
(see Figure 2). Since the elements of P are mutually at distance at least 4 we obtain that
N (P) R = q P R = (q 1) P . Hence by (2)
| Gq \ | | |−| | − | |
N3 (((cid:37),1,0) ) N (P) R + K = (q 1) P + K = (q 1)q(q 3)+ K .
| Gq 0 | ≥ | Gq \ | | | − | | | | − − | |
Since N3 (((cid:37),1,0) ) = q(q 1)2 we obtain that K 2q(q 1) yielding by (3) that N (R)
| Gq 0 | − | | ≤ − | Gq \
P q(q 1). As P contains at least q elements mutually at distance 6, R contains at least q
| ≤ −
elements mutually at distance 4. Thus we have N (R) P q2 q. Therefore N (R) P =
| Gq \ | ≥ − | Gq \ |
q2 q and all the above inequalities are actually equalities. Thus by (2) we get
−
N (R) = q2 q+ P = 2q(q 2) (4)
| Gq | − | | −
5
R
(̺,j,k)0
(j,t,k)1 P
(̺,1,k′)0
(̺,1,0)0 (1,b,0)1 (w,w+b,w+2b)0 NG (R)
q
(̺,1,k)0
NGq((̺,1,0)0) NG5q((̺,1,0)0)
NG2q((̺,1,0)0) NG4q((̺,1,0)0)
NG3q((̺,1,0)0)
Figure 1: Structure of the graph G . The eliminated vertices are inside the
q
Figure 2: Structure of the graph G . The perfect dominating set lies inside the dotted box.
dotted box. q
and every v N (R) P has exactly 1 neighbor in N5 (((cid:37),1,0) ) R. Therefore we have
∈ Gq \ Gq 0 \
(cid:91)
N4 (((cid:37),1,0) ) N (R) = (N2 (((cid:37),1,j) ) P) N (R)
| Gq 0 \ Gq | | Gq 0 ∪ \ Gq |
j∈Fq\{0}
= q(q 1)2+q(q 3) 2q(q 2)
− − − −
= q(q 1)(q 2).
− −
Let us denote by E[A,B] the set of edges between any two sets of vertices A and B. Then
E[N3 (((cid:37),1,0) ),N4 (((cid:37),1,0) )] = q(q 1)3 and E[N3 (((cid:37),1,0) ),N4 (((cid:37),1,0) ) N (R)] =
| Gq 0 Gq 0 | − | Gq 0 Gq 0 \ Gq |
q(q 1)2(q 2). Therefore,
− −
E[N3 (((cid:37),1,0) ),N (R)] = q(q 1)3 q(q 1)2(q 2) = q(q 1)2 = N3 (((cid:37),1,0) ) ,
| Gq 0 Gq | − − − − − | Gq 0 |
which implies that every v N3 (((cid:37),1,0) ) has exactly one neighbor in N (R). It follows that
∈ Gq 0 Gq
(cid:83)
S = N [((cid:37),1,j) ] N [R] is a perfect dominating set of G . Furthermore, by (2) and
j∈Fq Gq 0 ∪ Gq q
(4), S = q2 +q +q(3q 7) = 4q2 6q. Therefore a (q 1)-regular graph of girth 8 can be
| | − − −
obtained by deleting from G the perfect dominating set S, see Figure 2. This graph has order
q
2q(q2 2) 2q(2q 3) = 2q(q 1)2.
− − − −
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