Biregular cages of girth five M. Abreu1 ∗ G. Araujo-Pardo2 ∗ C. Balbuena3 ∗ D. Labbate1 ∗ G. L´opez-Ch´avez2 ∗ 1 Dipartimento di Matematica, Informatica ed Economia Universit`adegli Studi della Basilicata, Viale dell’Ateneo Lucano, I-85100 Potenza, Italy [email protected]; [email protected] 2 Instituto de Matem´aticas Universidad Nacional Aut´onoma de M´exico Ciudad Universitaria, M´exico D.F. 04510, M´exico. [email protected]; [email protected] 3 Departament de Matem`atica Aplicada III Universitat Polit`ecnica de Catalunya Campus Nord, Edifici C2, C/ Jordi Girona 1 i 3 E-08034 Barcelona, Spain. [email protected] Submitted: Aug 1, 2012; Accepted: Mar 25, 2013; Published: Mar 31, 2013 Mathematics Subject Classifications: 05C35 Abstract Let 2 6 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 paper we generalize the reduction and graph amalgam operations from [M. Abreu, G. Araujo–Pardo, C. Balbuena, D. Labbate. Families of Small Reg- ular Graphs of Girth 5. Discrete Math. 312(18) (2012) 2832–2842] on the inci- dence graphs of an affine and a biaffine plane obtaining two new infinite families of biregular cages and two new semiregular cages. The constructed new 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. Keywords: biregular, cage, girth. ∗Research supported by the Ministerio de Educaci´on y Ciencia, Spain, and the European Regional Development Fund (ERDF) under project MTM2011-28800-C02-02, CONACyT-M´exico under projects 166306 and 178395 and PAPIIT-M´exico under project IN101912. the electronic journal of combinatorics 20(1) (2013), #P71 1 1 Introduction All graphs considered are finite, undirected and simple (without loops or multiple edges). For definitions and notations not explicitly stated the reader may refer to [15], [13] and [16]. 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 length g = g(G) of a shortest cycle. The degree of a vertex v ∈ V is the number of vertices adjacent to v. A graph is called r–regular if all its vertices have the same degree r. A(r, g)–graph is a r–regular graph of girth g and a (r, g)–cage is a (r, g)–graph with the smallest possible number of vertices. Cages have been intensely studied since they were introduced by Tutte [31] in 1947. Erd˝os and Sachs [20] proved the existence of a (r, g)–graph for any value of r and g. Biggs is the author of an impressive report on distinct methods for constructing cubic cages [14]. More details about constructions of cages can be found in the surveys by Wong [32], by Holton and Sheehan [27, Chapter 6], or the recent one by Exoo and Jajcay [21]. The cages theory has been generalized in many ways, one such is as follows: if D = {a1,...,ak} is a set of positive integers with 2 6 a1 <a2 <...<ak then a (D; g)–graph is a graph with degree set D and girth g and a (D; g)–cage is a (D; g)–graph with minimum order n(D; g)= n(a1,...,ak; g). It is obvious that the (r; g)–cage is a special case of the (D; g)–cage when D = {r}. Few values of n(D; g) are known. In particular, Kapoor et al. [28] proved that n(D;3)=1+ ak. Moreover, the following lower bound for n(D; g) was given by Downs et al. [18]: t i−1 1+ ak(a − 1) if g =2t + 1; 1 > i=1 n(D; g) t−1 (1) 1+ a (a − 1)i−1 +(a − 1)t−1 if g =2t. k 1 1 i=1 A biregular ({r, m}; g)–graph isa(D; g)–graph with degree set D = {r, m} and girth g and a bi-regular ({r, m}; g)–cage is an ({r, m}; g)–graph of smallest possible order. Note that bouquets of r cycles of length g are ({2, 2r}; g)-cages and the complete bipartite graphs Kr,m are ({r, m};4)–cages. If m = r +1, an ({r, m}; g)–cage is said to be a semiregular cage. The existence of biregular ({r, m}; g)–graphs has been proved by Chartrand, Gould, and Kapoor in [17] for all 2 6 r <m and g > 3 (also proved by F¨uredi et al. in [23]). On the other hand, several contructions of biregular ({r, m}; g)–cages have been achieved for different values of r, m and g. In particular, Chartrand et al. proved in [17] that n({r, m};4) = r + m, for 2 6 r<m, and g = 4 and they also proved in [17] that n({2,m}; g) attains the lower bound (1). Furthermore, Yuansheng and Liang in [33] proved that n({r, m};6) > 2(rm − m +1) for any 2 6 r<m; that n({r, m};6) = 2(rm − m +1) for g = 6 and r<m when 2 6 r 6 5 or r > 2 and m − 1 a prime power; and they conjectured that n({r, m};6) = 2(rm − m +1), for any 2 6 r<m. the electronic journal of combinatorics 20(1) (2013), #P71 2 In this paper, we focus our attention on biregular graphs of girth exactly five, where n({r, m};5) = rm+1. Some known results in this case, Downs et al. [18] have shown that n({3; m};5)=3m +1, for any m > 4, Hanson et al. [24] have shown that n({4; m};5) = 4m+1, for any integer m > 5, and Araujo–Pardo et al. in [7] proved several results in this context. Moreover, it is worth to note that Yuansheng and Liang in [33], claim to have proved that n({5,m};5) = 5m +1 for m > 6 in an unpublished manuscript. In Table 1, we summarize these and other known results on the exact values of n({r, m}; g), for g > 5, together with the results obtained in this paper which we now proceed to describe. r m g = 5 g = 6 g = 7 g = 8 g = 9 g = 11 25m r = 3 m > 4 3m + 1 4m + 2 7m + 1 + 5 15m + 1 31m + 1 3 [18] [24, 33] [18] m = 3k [7] m > 6 m = 4k 9m + 3 [18] [7] m = 4, 5, 7 [10] r = 4 m > 5 4m + 1 6m + 2 13m + 1 121m + 1 [24] [33] m = 6k m = 6k [7] [7] − 5− r > m k r rm m r2 r m (r 1) 1 5 = 2 ( − 1) 1+ 1+ ( − + 1) 1+ r−2 r = ph + 1 [7] [7] [7] p prime m = k(r − 1) + 1 2(rm − m + 1) [6, 33] m = kr [7] m = 2r − 3 (*) h = 1 m = 2r − 5 (*) p > 7 Table 1: Exact values of n({r, m}; g). The symbol (*) means results obtained in this paper. We generalize the reduction and graph amalgam operations from [1] on the incidence graphs Aq of an affine and Bq of a biaffine plane (elliptic semiplane of type C) 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 paper is organized as follows: the graphs Aq and Bq are presented in Section 2, with a labelling which will be necessary for the construction of biregular cages. In Section 3, we construct previously unknown ({r, 2r−3};5)–cages, for a prime power q > 2 and the integer r = q +1, in Theorem 3.3, adding edges to the graph Aq. Moreover if r −1 is even, we exhibit r non-isomorphic such ({r, 2r − 3};5)–cages in Theorem 3.5. In particular, we find a ({4, 5};5)–semiregular cage with 21 vertices. In Section 4 we slightly generalize reduction and amalgam operations, described in [1] and [22], that, performed on the bipartite graph Bq, will allow us to construct new ({r, 2r −5};5)–cages, for r = q +1 with q > 7 prime, in Section 5, Theorems 5.10 and 5.11. Finally, in Section 6 we construct two new semiregular cages, namely a ({5, 6};5)-cage with 31 vertices and a ({6, 7};5)- cage with 43 vertices. Note that the latter is a sporadic example in which we adapt and slightly generalize the techniques that we have used in Section 4. the electronic journal of combinatorics 20(1) (2013), #P71 3 2 Preliminaries Let q = pn > 2 be a prime power and α a primitive (q − 1)th–root of unity. Consider the finite field GF (q)= {0, 1,α,α2,...,αq−2} and denote GF ∗(q)= GF (q) \ {0}.