ON HAMILTONIAN REGULAR GRAPHS OF GIRTH SIX. WILLIAM G. BROWN. 1. Introduction. In this paper we shall determine, when 1 = 6, bounds for numbers.
ON HAMILTONIAN REGULAR GRAPHS OF GIRTH SIX WILLIAM G. BROWN
1. Introduction In this paper we shall determine, when 1 = 6, bounds for numbers f(k, I) and F{k, 1) defined as follows: f{k, l)/F(k, I) is defined to be the smallest integer n for which there exists a regular graph/Hamiltonian regular graph of valency k and girth I having n vertices. The problem of determining minimal regular graphs of given girth was first considered by Tutte [9]. Bounds for f(k, I) have been obtained by Erdos and Sachs [2], while certain values of F(K, 6) have been found by Karteszi [6]. We shall determine an improved upper bound for f(k, 6) and also an upper bound for F(k, 6); our results will be best possible, in an asymptotic sense. In our constructions we shall utilize elementary properties of finite projective planes, and properties of the distribution of primes. The author wishes to acknowledge the assistance of the following of his colleagues during the course of this research: Professors W. McWorter, Z. A. Melzak, R. Westwick, and G. K. White; he is indebted to the referee for suggested simplifications in §6. 2. Preliminary definitions We are concerned with finite undirected graphs without loops or multiple edges; such graphs will be denoted by capital Gothic letters, as ©. If © has n vertices we may indicate this by a superscript, as ©n. ©n is said to be regular of valency k if each of its vertices has valency k; the graph may then be denoted by &©n or &©. If vertices v and w are connected by an edge in ©, that edge will be denoted by (v, w) or (w, v). A circuit P=((v1, ..., vs)) in © is a cyclic sequence (unique up to reversal) of distinct vertices vi such that edges {vi} vi+1) exist in © {i = 1, 2,..., s; s > 0; indices modulo s); such a circuit P will be said to be an s-gon and to have length s. A Hamiltonian circuit in (Bn is defined to be an w-gon in ©n, i.e., a circuit containing all vertices of ©; © is Hamiltonian if it possesses a Hamiltonian circuit. The girth of a graph ©, which we shall denote by y(©), is defined to be the infimum of the lengths of all circuits in ©. f{k, I) = def inf {n 13© = &©« such that y (©) = I}, F(k, 1) = def inf {n \ 3© = fc©n such that y(©) = I and © is Hamiltonian}. Received 19 January, 1965; revised 6 March, 1966. [JOTTRNAI LONDON MATH. SOO., 42 (1967), 514-520]
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3. Summary of known results for 1 = 6 (We shall always assume k > 3. / ( I , 6) and -F(l, 6) are infinite; clearly /(2, 6) = JF(2,6) = 6; /(3, 6) = J ( 3 , 6) = H (c/.[9]». In [2] Erdos and Sachs proved that for all k (3.1)
2(Jfc*-fc+lK/(fc,6), /(Jfc,6) 0 there exists an integer Ne such that for all k > Ne, 2
From (3.1) and (3.4) we obtain the (3.5)
COROLLARY.
lim/(&, Q)jk2= \im F(k, 4. The Levi graph of a finite projective plane Given a finite projective plane TT (not necessarily Desarguesian) of order t {i.e., whose points/lines are each incident with t+l lines/points) we construct its Levi graph (cf. [1]) which we denote by £(TT), as follows. The vertices of our graph, 2(t2 +1 + 1) in number, represent the points and lines of TT ; two vertices are connected by an edge if and only if they represent a point and line which are incident. The axioms of the geometry ensure that the graph has girth 6 and is regular of valency t + 1 [1; p. 425; 6 ; 8]. It can also be shown [8] that any graph *@2(*a-*+i) of girth 6 can be interpreted as the Levi graph of some finite projective plane (of order k — 1); thus the Bruck-Ryser theorem [4; p. 394] can be applied to improve (3.1) slightly in some cases. We can label the vertices of 2(TT) in the following way. Select in TT a point P o and a line l0 incident with it. Let lv ..., lt be the other lines incident
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with P o ; and Pls ..., Pt the other points incident with l0. We shall denote the class of points/lines incident with li(Pi by [JJ/[PJ {i = 0, 1, ...,t). Points and lines will be identified with the vertices representing them in the Levi graph. t
t
(4.1) LEMMA. Each vertex in U [Pt] - {/0}/ U [lt] — {Po} is connected il
1
to exactly one vertex in each of IJPj 0 = 1, • • •, t). We remark finally, without proof, that non-isomorphic planes of order t have non-isomorphic Levi graphs. Thus, as non-isomorphic geometries are known to exist for certain orders, the minimal regular graphs of girth 6 need not be unique, even when equality holds in (3.1). 5. Construction of a family of regular graphs of girth 6 Let 77 be a plane of order t ^ 3, and let £(IT) be its Levi graph; (f£ 3 as we assumed k ^ 4). Let r be an integer such that (5.1)
O^r^t-S.
Suppose the points and lines of IT labelled as described in §4; (later we shall impose further conditions on this labelling). We denote by ^(n) the r
r
graph obtained by erasing from £(TT) the vertices in U (7J and U [PJ and i=0
all incident edges. (5.2) COROLLARY TO (4.1).
rZ(ir)
i=0
is regular of valency t — r.
Evidently y{r%{n)j ^ 6. However, it is not generally true that r£(7r) has girth 6. (It is helpful to consider a geometric interpretation of the preceding erasures. rS (TT) is the Levi graph of a configuration in TT consisting of all points (^ Po) on t — r lines (# l0) through P o , and all lines (# lQ) through t — r points (^Po) on l0. This configuration need not contain three non-concurrent lines which intersect in pairs. Nevertheless, there must exist in IT three non-concurrent lines: these determine a 6-gon in the Levi graph. We shall assume henceforth that the points and lines are so labelled that some such 6-gon involves no vertex in t
[P0]w[Z0]; and, moreover, involves vertices in
U ([P^]^[^]) only.
(The proof that such restrictions are possible we leave as an exercise for the reader.) (5.3) THEOREM. For every integer k> 3 and every prime power q= (more generally, perhaps, for every integer q^k for which there exists a finite protective plane of order q),f{k, 6) ^ 2qk. Proof. Let re be a finite plane of order q ^ k. Then, in a notation conforming to (5.3), q-k%(n) has 2qk vertices and girth 6, and its vertices have valency k by (6.2).
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6. Construction of a family of Hamiltonian regular graphs of girth 6 We shall prove that there exists a labelling of every finite Desarguesian geometry 7T = PG(2,p) of prime order p>2 for which every ^(TT) is Hamiltonian {O^r^p — 3). (Our proof cannot be generalized to prime powers q as the construction involves implicity a collineation with the following properties: (i) s has exactly one fixed point Po and exactly one fixed line l0 incident with P o (s = 1, 2,..., q — 1); (ii) ^ r = i . It can be shown that such collineations exist only for prime q.) We shall select a coordinate system in PG(2, p) as follows: points/lines will be represented by ordered triples (xQ, xx, x2)l[y0, yv y2] of elements of GF(p) not all zero, unique up to constant multiples; (x0, xv x2) and [yQ, yv y2] will be incident if and only if (6.1) Moreover, we shall assume the coordinate system so chosen that PQ = (0, 0, 1), Zo= (1, 0, 0], and P*= (0, 1, -»), ! 2 there exists a prime p such that kNe there exists a primep such that k
