Counting irregular multigraphs

Gagliardi et al. (1996, unpublished manuscript) defined an irregular multigraph to be a loopless multigraph with degree sequence n, n 1 . . . . . 1, and they posed the problem of determining the number of different irregular multigraphs f , on n vertices. In Gagliardi et al. (1996) they showed that if n 0 or 3 (mod 4) then f , > n 1. In this note our aim is to show that there are constants 1 < cl < c2 and no > 0 such that if n/> no and n = 0 or 3 (mod 4) then (cl).2 < f , < (c2),2. Indeed, we show that c~ = 1.19 and c2 = 1.65 can be chosen. (~) 1999 Elsevier Science B.V. All rights reserved In this note we consider loopless multigraphs. V(G) denotes the vertex set, E(G) denotes the edge set of the multigraph G. For two multigraphs G and H, the union of G and H, written GUH, has vertex set V(G)U V(H) and edge set E(G)UE(H) . Gagliardi et al. [1,2] defined an irregular multigraph to be a loopless multigraph with degree sequence n, n 1 . . . . . 1, and they posed the problem of determining the number of different irregular multigraphs on n vertices. We define f , to be the number of irregular multigraphs on n vertices. As Gagliardi et al. [1] show, the even parity of ~-~'~l ~i.<, di clearly implies that n ~ 0 or 3 (mod4) is a necessary condition for f , >0 . They also established that if n 0 or 3 (mod4) , then f , > n 1. Our goal is to provide the following bounds for f , . Note that 1 f . = [zg . . . . . z,] 11( + z ,z#, = [z." . . . . . I [ i -z zj is~j i•j * Corresponding author. E-mail: [email protected]. 0012-365X/99/$-see front matter (~) 1999 Elsevier Science B.V. All rights reserved PII: S0012-365X(98)001 90-3 236 A.C. Atkins et al./Discrete Mathematics 195 (1999) 235 237 Theorem 1. There are constants 1 0 such that if n >~no and n 0 or 3(mod4) then (cl) ~2 < f , < ( c 2 ) #'. Indeed, we show that cl = 1.19 and c2 = 1.65 can be chosen. Proof. For the lower bound, we partition the n vertices into sets A and B of sizes d and n d respectively, with d = d(n) = 2 ( n 1 ) / ( 2 + 1), where 2 is a constant to be chosen later with the property that d is divisible by 4. We place fixed irregular multigraphs on A and B (these exist since 41d and n d = 0 or 3 (mod4)). If we superimpose on B any d-regular multigraph B*, then A tO B tO B* is an irregular multigraph on n vertices. Since any d-regular multigraph B* superimposed on B yields a unique irregular multigraph AUBUB*, the number of d-regular (labeled) graphs on n d vertices is a lower bound on fn. A deep result of McKay and Wormald [3] implies that given the above conditions on d, as n d ~ cx~ the number of (labeled) d-regular graphs is at least C (2n(n d),~d+l(1 -2) n-2a)("-a)/2' where C is an absolute constant. Thus, we get 1 C (2a+l(1 2)n--2a)Cn--a)/2 (2rffn -d))Cn-a)/2 >/(2-~/2~;'+1)2(1 2) (4-1)/2(';~+1)2 ) n2 C (2~t(n d) ) (n-d)~2" Choosing 2 to be the largest real number that is at most 0.3 and for which d is divisible by 4, we get that if n is sufficiently large, then f . >(1.19) "2. To establish the upper bound, we note that if for some 1 ~<k ~<n 1, vk has j neighbors among {vk+l . . . . . v.} (where 0 ~<j ~<k), then the number of ways to distribute these j edges among {Vk+l . . . . . V.} is bounded from above by the number of ways to sample, with replacement, j elements from n k elements, or ( n k +j-1).j Thus an upper bound on the number of ways to distribute the edges is L<. n (x (:) (:) l ~ k ~ n I l ~ k ~ n I l ~ k ~ n ~ n k k ( n _ k ) . k = e x p og n k k ( , 2 k ) ~ k l ~ k ~ n l ~ k ~ n = exp Z log kk(n Z-k)"-k 1 <~k<~n A. C A tkins et al. I Discrete Mathematics 195 (1999) 235-237 237 ~exp(n21ogn-2fo~xlogxdx) ox (9 where x/~ ~ 1.64872. [] =exp(n21ogn-2[ x21°gx X2] n) 2 4 0 We note, that for the lower bound of the previous theorem, we can allow A to be any irregular multigraph and B to be a set of irregular multigraphs such that if we superimpose any d-regular graph on distinct members of B it is impossible to generate an irregular multigraph in two different ways. Using this idea, we can raise the value of Cl to be roughly 1.3. We omit the details, since the computation is somewhat tedious and the result is still far from the upper bound. The determination of the asymptotic growth fn remains an open problem. We conjecture that if n = 0 or 3 (mod4), then l i m , ~ ( f n ) b'''2 = x/2.