- Email: [email protected]
Constructions of biregular cages of girth five
Available online at www.sciencedirect.com
Electronic Notes in Discrete Mathematics 40 (2013) 9–14 www.elsevier.com/locate/endm
Constructions of biregular cages of girth five M. Abreu a,1 a
Dipartimento di Matematica Informatica ed Economia, Universit` a della Basilicata, Potenza, Italy
G. Araujo-Pardo b,1 C. Balbuena c,1 D. Labbate G. L´opez-Ch´avez b,1
a,1
b
Instituto de Matem´ aticas, Universidad Nacional Aut´ onoma de M´exico, M´exico D.F., M´exico c
Departament de Matem` atica Aplicada III Universitat Polit`ecnica de Catalunya, Barcelona, Spain
Abstract Let 2 ≤ r < m and g be positive integers. An ({r, m}; g)–graph (or biregular graph) is a graph with degree set {r, m} and girth g, and an ({r, m}; g)–cage (or biregular cage) is an ({r, m}; g)–graph of minimum order n({r, m}; g). If m = r + 1, an ({r, m}; g)–cage is said to be a semiregular cage. In this extended abstract we construct two infinite families of biregular cages and two semiregular cages, obtained from the incidence graphs of an affine and a biaffine plane by generalizations of the reduction and graph amalgam operations from [1]. Keywords: biregular, cage, girth 1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.05.003
10
1
M. Abreu et al. / Electronic Notes in Discrete Mathematics 40 (2013) 9–14
Introduction
For graph definitions and notations not explicitly stated the reader may refer to [7]. Let 2 ≤ r < m and g be positive integers. An ({r, m}; g)–graph (or biregular graph) is a graph with degree set {r, m} and girth g, and an ({r, m}; g)–cage (or biregular cage) is an ({r, m}; g)–graph of minimum order n({r, m}; g). If m = r + 1, an ({r, m}; g)–cage is said to be a semiregular cage. These graphs correspond to one of the possible generalizations of cages, which have been intensely studied since they were introduced by Tutte [14] in 1947. For the most recent survey in this topic refer to Exoo and Jajcay [10]. A lower bound for n({r, m}; g) was given, in a more general context, by Downs et al. [9]. The existence of biregular ({r, m}; g)–graphs has been proved by Chartrand et al. in [8] for all 2 ≤ r < m and g ≥ 3. On the other hand, several contructions of biregular ({r, m}; g)–cages have been achieved for different values of r, m and g. Here, we focus our attention on biregular graphs of girth exactly five, where n({r, m}; 5) = rm+1 from [9]. Some known results in this case can be found in [9,12] and [5]. In this extended abstract (based on [2]), we generalize the reduction and graph amalgam operations from [1,11] and [13] on the incidence graphs Aq of an affine and Bq of a biaffine plane obtaining two new infinite families of biregular ({r, m}; 5)–cages and two new semiregular cages of girth 5. The constructed families are ({r, 2r − 3}; 5)–cages for all r = q + 1 with q a prime power, and ({r, 2r − 5}; 5)–cages for all r = q + 1 with q a prime. The new semiregular cages are constructed for r = 5 and 6 with 31 and 43 vertices respectively. The following labelling on V (Bq ) and V (Aq ) is central for our constructions: Definition 1.1 Let q ≥ 2 be a prime power and GF (q) a finite field. (i) Let Bq be a bipartite graph with vertex set (V0 , V1 ) where Vr = GF (q) × GF (q), r = 0, 1; and the edge set defined as follows: (1)
(x, y)0 ∈ V0 adjacent to (m, b)1 ∈ V1 if and only if y = mx + b.
(ii) Let Aq be the graph obtained from Bq by adding the following set Lq := {(q, x)1 | x ∈ GF (q)} of q vertices and the set Eq := {uv | u := (q, x)1 , v := (x, y)0 and x, y ∈ GF (q)} of q 2 edges. The graphs Bq and Aq admit a further partition of their vertex set, i.e. V0 = 1
Email: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]
M. Abreu et al. / Electronic Notes in Discrete Mathematics 40 (2013) 9–14
[
Px and V1 =
x∈GF (q)
[
11
Lm where Px = {(x, y)0 | y ∈ GF (q)}, x ∈ GF (q),
m∈GF (q)
and Lm = {(m, b)1 | b ∈ GF (q)}, m ∈ GF (q) (m ∈ GF (q) ∪ {q} in Aq ). The graph Bq has been widely used in the problem of finding extremal graphs without short cycles (cf. e.g. [1,3,4,6]).
2
Construction of a family of ({r, 2r − 3}; 5)–cages.
Let q ≥ 2 be a prime power and let r = q + 1. We define Rq to be the graph with V (Rq ) := V (Aq ) and E(Rq ) := E(Aq )∪D where D = {(m, 0)1 (m, b)1 | b ∈ GF ∗ (q) and m ∈ GF (q) ∪ {q}}. Theorem 2.1 Let q ≥ 2 be a prime power and let r = q + 1. Then the graph Rq is a ({r, 2r − 3}; 5)–cage of order r(2r − 3) + 1. ✷ Corollary 2.2 The graph Rq is a semi–regular cage if and only if r = 4. ✷ Similarly, for q even, adding matchings instead of stars within some of the blocks Lm we have: Theorem 2.3 Let q = 2s be an even prime power, with s ≥ 1. Then there are at least q + 1 non–isomorphic ({r, 2r − 3}; 5)–cages. ✷
3
Operations on Bq
Reduction 1[1]
Remove vertices from P0 and L0 . Let T ⊆ S ⊆ GF (q), S0 = {(0, y)0 |y ∈ S} ⊆ P0 , T0 = {(0, b)1 |b ∈ T } ⊆ L0 and Bq (S, T ) = Bq − S0 − T0 . Lemma 3.1 Let T ⊆ S ⊆ GF (q). Then Bq (S, T ) is biregular with degrees (q − 1, q) and order 2q 2 − |S| − |T |. Remove blocks Pi and Lj from Bq or from Bq (S, T ). Let u0 , u1 be non–negative integers such that 0 ≤ u0 ≤ u1 < q − 1. We define Bq (u0 , u1 ) to be the graph obtained from Bq by deleting the last u0 blocks of V0 , and the last u1 blocks of V1 . Analogously, we define Bq (S, T, u0 , u1 ) to be a similar graph obtained from the graph Bq (S, T ) (c.f. Reduction 1). Clearly, for u0 = u1 = 0, Bq (0, 0) = Bq and Bq (S, T, 0, 0) = Bq (S, T ). Reduction 2
Lemma 3.2 Let u0 , u1 be non–negative integers, with 0 ≤ u0 ≤ u1 < q − 1. Then the graph Bq (S, T, u0 , u1 ) has degrees {q −u0 , q −u1 , q −u0 −1, q −u1 −1} and order 2q 2 − q(u0 + u1 ) − |S| − |T |.
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M. Abreu et al. / Electronic Notes in Discrete Mathematics 40 (2013) 9–14
Add edges to Bq (S, T, u0 , u1 ) to achieve desired degrees. Let Γ1 and Γ2 be two graphs of the same order and with the same labels on their vertices. In general, an amalgam of Γ1 into Γ2 is a graph obtained adding all the edges of Γ1 to Γ2 . Consider the graph Bq (S, T, u0 , u1 ). Let H1 , H2 , Gi , for i = 1, 2, be graphs of girth at least 5 and appropriate orders. We define Bq∗ (S, T, u0 , u1 ) to be the amalgam of H1 into P0′ , H2 into L′0 , G1 into Pi , for i 6= 0, and G2 into Lj , for j 6= 0. Note that |V (Bq∗ (S, T, u0 , u1 ))| = |V (Bq (S, T, u0 , u1 ))|. Let F ∈ {H1 , H2 , G1 , G2 } and let MF := {(u, v) : u, v ∈ GF (q) and uv ∈ E(F )}. For each (u, v) ∈ MF , we define ω((u, v)) = ±(u − v) ∈ GF ∗ (q) to be its weight and Ω(F ) := {ω((u, v)) : (u, v) ∈ MF } to be the set of weights of F . The following lemma generalizes [1, Theorem 5] and [11, Theorem 2.8]. Amalgam
Theorem 3.3 Let T ⊆ S ⊆ GF (q) and let 0 ≤ u0 ≤ u1 < q − 1. Let H1 , H2 , G1 and G2 be defined as above and suppose that MH1 ∩ MH2 = ∅, MH1 ∩ MG2 = ∅, MH2 ∩ MG1 = ∅ and Ω(G1 ) ∩ Ω(G2 ) = ∅. Then the amalgam Bq∗ (S, T, u0 , u1 ) has girth at least 5 and order 2q 2 − q(u0 + u1 ) − |S| − |T |.
4
New ({r, 2r − 5}; 5)-cages for r ≥ 8 and New semiregular ({r, r + 1}; 5)-cages for r = 5, 6.
3q−1 For primes q = 4n + 3, consider S = { q+1 , − q+1 } = { q+1 , }; T = ∅; 4 4 4 4 n ≥ 1 q−1 q+1 3q−1 3q+3 3q+3 q−3 u0 = 0; u1 = 1; H1 = (j, j + 2 ) | j ∈ Zq − { 4 , 4 , 4 } ∪ ( 4 , 4 ) (a (q − 2)–cycle with Ω(H1 ) = { q−1 , q−3 }), G1= (j, j + q−1 ) | j ∈ Zq (a q– 2 2 2 q−1 ∗ ∼ cycle with Ω(G1 ) = { 2 }) and H2 = G2 = (0, j) : j ∈ Zq − { q−1 , q+1 } ∪ 2 2 q+1 q−1 3q−1 q+1 ( 4 , 2 ), ( 4 , 2 )} (with Ω(H2 ) = Ω(G2 ) = Z∗q − { q−1 }). 2
Theorem 4.1 Let q = 4n+3 be a prime, for n ≥ 1. Let S, T, u0 , u1 , H1 , H2 , G1 and G2 be defined as above. Then the amalgam graph Bq∗ (S, T, u0 , u1 ) is an ({r, 2r − 5}; 5)–cage of order r(2r − 5) + 1, where r = q + 1. Similarly, for primes q = 4n + 1, n ≥ 3 we have: Theorem 4.2 Let q = 4n+1 be a prime, for n ≥ 3. Let S, T, u0 , u1 , H1 , H2 , G1 and G2 be defined similarly as before. Then the amalgam graph Bq∗ (S, T, u0 , u1 ) is an ({r, 2r − 5}; 5)–cage of order r(2r − 5) + 1, where r = q + 1. Theorem 4.3 Let q = 4, let GF (4) = {0, 1, α, α2 } be the finite field of order 4 and let S = {0}, T = ∅, u0 = u1 = 0, H1 = {(1, α2 ); (α2 , α)}, G1 = {(0, α); (1, α2 )} and H2 = G2 = {(α, 1); (1, 0); (0, α2 )}. Then the amalgam graph B4∗ (S, T, u0 , u1 ) is a ({5, 6}; 5)–cage of order 31.
M. Abreu et al. / Electronic Notes in Discrete Mathematics 40 (2013) 9–14
13
For the last graph we consider a slightly modified version of Reduction 1 and the corresponding amalgam. Let T := {3} ⊂ GF (5), S := {0, 3} ⊂ GF (5), S0 = {(0, y)0 |y ∈ S} ⊆ P[ 0 , Tj = {(j, b)1 |b ∈ T } ⊆ Lj for j ∈ GF (5), and let B5 (S, T T ) := B5 − S0 − Tj . Consider H1 = {(1, 4); (4, 2)}, G1 = j∈GF (5)
{(0, 2); (2, 4); (4, 1); (1, 3); (3, 0)}, H2 = {(2, 1); (1, 0); (0, 4)} and let B5∗ (S, T T ) be the graph obtained from the amalgam of H1 into P0′ := P0 − S0 , G1 into Pi , for all i ∈ GF ∗ (5), and H2 into L′j = Lj − Tj , for all j ∈ GF (5). Theorem 4.4 Let q = 5 and let S, T, H1 , H2 and G1 be defined as above. Then the graph B5∗ (S, T T ) is a ({6, 7}; 5)–cage of order 43.
References [1] M. Abreu, G. Araujo–Pardo, C. Balbuena, D. Labbate. Families of Small Regular Graphs of Girth 5. Discrete Math. 312(18) (2012) 2832–2842. [2] M. Abreu, G. Araujo–Pardo, C. Balbuena, D. Labbate, G. L´opez–Ch´avez. Biregular cages of girth five. (submitted) [3] M. Abreu, M. Funk, D. Labbate, V. Napolitano. On (minimal) regular graphs of girth 6. Australas. J. Combin. 35 (2006) 119–132. [4] M. Abreu, M. Funk, D. Labbate, V. Napolitano. A family of regular graphs of girth 5. Discrete Math. 308(10) (2008) 1810–1815. [5] G. Araujo-Pardo, C. Balbuena, J.C. Valenzuela, Constructions of bi-regular cages, Discrete Math. 309 (2009) 1409–1416. [6] G. Araujo-Pardo, C. Balbuena, Constructions of small regular bipartite graphs of girth 6. Networks 57(2) (2011) 121–127. [7] G. Chartrand, L. Lesniak Graphs and Digraphs, Chapman and Hall, 3rd edition, 1996. [8] G. Chartrand, R.J. Gould, S.F. Kapoor, Graphs with prescribed degree set and girth, Period. Math. Hungar., 6 (1981) 261–266. [9] M. Downs, R.J. Gould, J. Mitchem, F. Saba, (D; n)-cages, Congr. Numer. 32 (1981) 179–193. [10] G. Exoo and R. Jajcay, Dynamic Cage Survey, Electron. J. Combin. 15 (2008) #DS16. [11] M. Funk, Girth 5 graphs from elliptic semiplanes, Note di Matematica, 29 suppl. 1 (2009) 91–114. [12] D. Hanson, P. Wang, L. Jorgensen, On cages with given degree sets, Discrete Math., 101 (1992) 109–114.
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[13] L. Jørgensen, Girth 5 graphs from difference sets, Discrete Math. 293 (2005) 177–184. [14] W. T. Tutte, A family of cubical graphs. Proc. Cambridge Philos. Soc., (1947) 459–474.
Electronic Notes in Discrete Mathematics 40 (2013) 9–14 www.elsevier.com/locate/endm
Constructions of biregular cages of girth five M. Abreu a,1 a
Dipartimento di Matematica Informatica ed Economia, Universit` a della Basilicata, Potenza, Italy
G. Araujo-Pardo b,1 C. Balbuena c,1 D. Labbate G. L´opez-Ch´avez b,1
a,1
b
Instituto de Matem´ aticas, Universidad Nacional Aut´ onoma de M´exico, M´exico D.F., M´exico c
Departament de Matem` atica Aplicada III Universitat Polit`ecnica de Catalunya, Barcelona, Spain
Abstract Let 2 ≤ r < m and g be positive integers. An ({r, m}; g)–graph (or biregular graph) is a graph with degree set {r, m} and girth g, and an ({r, m}; g)–cage (or biregular cage) is an ({r, m}; g)–graph of minimum order n({r, m}; g). If m = r + 1, an ({r, m}; g)–cage is said to be a semiregular cage. In this extended abstract we construct two infinite families of biregular cages and two semiregular cages, obtained from the incidence graphs of an affine and a biaffine plane by generalizations of the reduction and graph amalgam operations from [1]. Keywords: biregular, cage, girth 1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.05.003
10
1
M. Abreu et al. / Electronic Notes in Discrete Mathematics 40 (2013) 9–14
Introduction
For graph definitions and notations not explicitly stated the reader may refer to [7]. Let 2 ≤ r < m and g be positive integers. An ({r, m}; g)–graph (or biregular graph) is a graph with degree set {r, m} and girth g, and an ({r, m}; g)–cage (or biregular cage) is an ({r, m}; g)–graph of minimum order n({r, m}; g). If m = r + 1, an ({r, m}; g)–cage is said to be a semiregular cage. These graphs correspond to one of the possible generalizations of cages, which have been intensely studied since they were introduced by Tutte [14] in 1947. For the most recent survey in this topic refer to Exoo and Jajcay [10]. A lower bound for n({r, m}; g) was given, in a more general context, by Downs et al. [9]. The existence of biregular ({r, m}; g)–graphs has been proved by Chartrand et al. in [8] for all 2 ≤ r < m and g ≥ 3. On the other hand, several contructions of biregular ({r, m}; g)–cages have been achieved for different values of r, m and g. Here, we focus our attention on biregular graphs of girth exactly five, where n({r, m}; 5) = rm+1 from [9]. Some known results in this case can be found in [9,12] and [5]. In this extended abstract (based on [2]), we generalize the reduction and graph amalgam operations from [1,11] and [13] on the incidence graphs Aq of an affine and Bq of a biaffine plane obtaining two new infinite families of biregular ({r, m}; 5)–cages and two new semiregular cages of girth 5. The constructed families are ({r, 2r − 3}; 5)–cages for all r = q + 1 with q a prime power, and ({r, 2r − 5}; 5)–cages for all r = q + 1 with q a prime. The new semiregular cages are constructed for r = 5 and 6 with 31 and 43 vertices respectively. The following labelling on V (Bq ) and V (Aq ) is central for our constructions: Definition 1.1 Let q ≥ 2 be a prime power and GF (q) a finite field. (i) Let Bq be a bipartite graph with vertex set (V0 , V1 ) where Vr = GF (q) × GF (q), r = 0, 1; and the edge set defined as follows: (1)
(x, y)0 ∈ V0 adjacent to (m, b)1 ∈ V1 if and only if y = mx + b.
(ii) Let Aq be the graph obtained from Bq by adding the following set Lq := {(q, x)1 | x ∈ GF (q)} of q vertices and the set Eq := {uv | u := (q, x)1 , v := (x, y)0 and x, y ∈ GF (q)} of q 2 edges. The graphs Bq and Aq admit a further partition of their vertex set, i.e. V0 = 1
Email: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]
M. Abreu et al. / Electronic Notes in Discrete Mathematics 40 (2013) 9–14
[
Px and V1 =
x∈GF (q)
[
11
Lm where Px = {(x, y)0 | y ∈ GF (q)}, x ∈ GF (q),
m∈GF (q)
and Lm = {(m, b)1 | b ∈ GF (q)}, m ∈ GF (q) (m ∈ GF (q) ∪ {q} in Aq ). The graph Bq has been widely used in the problem of finding extremal graphs without short cycles (cf. e.g. [1,3,4,6]).
2
Construction of a family of ({r, 2r − 3}; 5)–cages.
Let q ≥ 2 be a prime power and let r = q + 1. We define Rq to be the graph with V (Rq ) := V (Aq ) and E(Rq ) := E(Aq )∪D where D = {(m, 0)1 (m, b)1 | b ∈ GF ∗ (q) and m ∈ GF (q) ∪ {q}}. Theorem 2.1 Let q ≥ 2 be a prime power and let r = q + 1. Then the graph Rq is a ({r, 2r − 3}; 5)–cage of order r(2r − 3) + 1. ✷ Corollary 2.2 The graph Rq is a semi–regular cage if and only if r = 4. ✷ Similarly, for q even, adding matchings instead of stars within some of the blocks Lm we have: Theorem 2.3 Let q = 2s be an even prime power, with s ≥ 1. Then there are at least q + 1 non–isomorphic ({r, 2r − 3}; 5)–cages. ✷
3
Operations on Bq
Reduction 1[1]
Remove vertices from P0 and L0 . Let T ⊆ S ⊆ GF (q), S0 = {(0, y)0 |y ∈ S} ⊆ P0 , T0 = {(0, b)1 |b ∈ T } ⊆ L0 and Bq (S, T ) = Bq − S0 − T0 . Lemma 3.1 Let T ⊆ S ⊆ GF (q). Then Bq (S, T ) is biregular with degrees (q − 1, q) and order 2q 2 − |S| − |T |. Remove blocks Pi and Lj from Bq or from Bq (S, T ). Let u0 , u1 be non–negative integers such that 0 ≤ u0 ≤ u1 < q − 1. We define Bq (u0 , u1 ) to be the graph obtained from Bq by deleting the last u0 blocks of V0 , and the last u1 blocks of V1 . Analogously, we define Bq (S, T, u0 , u1 ) to be a similar graph obtained from the graph Bq (S, T ) (c.f. Reduction 1). Clearly, for u0 = u1 = 0, Bq (0, 0) = Bq and Bq (S, T, 0, 0) = Bq (S, T ). Reduction 2
Lemma 3.2 Let u0 , u1 be non–negative integers, with 0 ≤ u0 ≤ u1 < q − 1. Then the graph Bq (S, T, u0 , u1 ) has degrees {q −u0 , q −u1 , q −u0 −1, q −u1 −1} and order 2q 2 − q(u0 + u1 ) − |S| − |T |.
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M. Abreu et al. / Electronic Notes in Discrete Mathematics 40 (2013) 9–14
Add edges to Bq (S, T, u0 , u1 ) to achieve desired degrees. Let Γ1 and Γ2 be two graphs of the same order and with the same labels on their vertices. In general, an amalgam of Γ1 into Γ2 is a graph obtained adding all the edges of Γ1 to Γ2 . Consider the graph Bq (S, T, u0 , u1 ). Let H1 , H2 , Gi , for i = 1, 2, be graphs of girth at least 5 and appropriate orders. We define Bq∗ (S, T, u0 , u1 ) to be the amalgam of H1 into P0′ , H2 into L′0 , G1 into Pi , for i 6= 0, and G2 into Lj , for j 6= 0. Note that |V (Bq∗ (S, T, u0 , u1 ))| = |V (Bq (S, T, u0 , u1 ))|. Let F ∈ {H1 , H2 , G1 , G2 } and let MF := {(u, v) : u, v ∈ GF (q) and uv ∈ E(F )}. For each (u, v) ∈ MF , we define ω((u, v)) = ±(u − v) ∈ GF ∗ (q) to be its weight and Ω(F ) := {ω((u, v)) : (u, v) ∈ MF } to be the set of weights of F . The following lemma generalizes [1, Theorem 5] and [11, Theorem 2.8]. Amalgam
Theorem 3.3 Let T ⊆ S ⊆ GF (q) and let 0 ≤ u0 ≤ u1 < q − 1. Let H1 , H2 , G1 and G2 be defined as above and suppose that MH1 ∩ MH2 = ∅, MH1 ∩ MG2 = ∅, MH2 ∩ MG1 = ∅ and Ω(G1 ) ∩ Ω(G2 ) = ∅. Then the amalgam Bq∗ (S, T, u0 , u1 ) has girth at least 5 and order 2q 2 − q(u0 + u1 ) − |S| − |T |.
4
New ({r, 2r − 5}; 5)-cages for r ≥ 8 and New semiregular ({r, r + 1}; 5)-cages for r = 5, 6.
3q−1 For primes q = 4n + 3, consider S = { q+1 , − q+1 } = { q+1 , }; T = ∅; 4 4 4 4 n ≥ 1 q−1 q+1 3q−1 3q+3 3q+3 q−3 u0 = 0; u1 = 1; H1 = (j, j + 2 ) | j ∈ Zq − { 4 , 4 , 4 } ∪ ( 4 , 4 ) (a (q − 2)–cycle with Ω(H1 ) = { q−1 , q−3 }), G1= (j, j + q−1 ) | j ∈ Zq (a q– 2 2 2 q−1 ∗ ∼ cycle with Ω(G1 ) = { 2 }) and H2 = G2 = (0, j) : j ∈ Zq − { q−1 , q+1 } ∪ 2 2 q+1 q−1 3q−1 q+1 ( 4 , 2 ), ( 4 , 2 )} (with Ω(H2 ) = Ω(G2 ) = Z∗q − { q−1 }). 2
Theorem 4.1 Let q = 4n+3 be a prime, for n ≥ 1. Let S, T, u0 , u1 , H1 , H2 , G1 and G2 be defined as above. Then the amalgam graph Bq∗ (S, T, u0 , u1 ) is an ({r, 2r − 5}; 5)–cage of order r(2r − 5) + 1, where r = q + 1. Similarly, for primes q = 4n + 1, n ≥ 3 we have: Theorem 4.2 Let q = 4n+1 be a prime, for n ≥ 3. Let S, T, u0 , u1 , H1 , H2 , G1 and G2 be defined similarly as before. Then the amalgam graph Bq∗ (S, T, u0 , u1 ) is an ({r, 2r − 5}; 5)–cage of order r(2r − 5) + 1, where r = q + 1. Theorem 4.3 Let q = 4, let GF (4) = {0, 1, α, α2 } be the finite field of order 4 and let S = {0}, T = ∅, u0 = u1 = 0, H1 = {(1, α2 ); (α2 , α)}, G1 = {(0, α); (1, α2 )} and H2 = G2 = {(α, 1); (1, 0); (0, α2 )}. Then the amalgam graph B4∗ (S, T, u0 , u1 ) is a ({5, 6}; 5)–cage of order 31.
M. Abreu et al. / Electronic Notes in Discrete Mathematics 40 (2013) 9–14
13
For the last graph we consider a slightly modified version of Reduction 1 and the corresponding amalgam. Let T := {3} ⊂ GF (5), S := {0, 3} ⊂ GF (5), S0 = {(0, y)0 |y ∈ S} ⊆ P[ 0 , Tj = {(j, b)1 |b ∈ T } ⊆ Lj for j ∈ GF (5), and let B5 (S, T T ) := B5 − S0 − Tj . Consider H1 = {(1, 4); (4, 2)}, G1 = j∈GF (5)
{(0, 2); (2, 4); (4, 1); (1, 3); (3, 0)}, H2 = {(2, 1); (1, 0); (0, 4)} and let B5∗ (S, T T ) be the graph obtained from the amalgam of H1 into P0′ := P0 − S0 , G1 into Pi , for all i ∈ GF ∗ (5), and H2 into L′j = Lj − Tj , for all j ∈ GF (5). Theorem 4.4 Let q = 5 and let S, T, H1 , H2 and G1 be defined as above. Then the graph B5∗ (S, T T ) is a ({6, 7}; 5)–cage of order 43.
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