Some Applications of Spanning Trees in K s,t

Size: px

Start display at page:

Download "Some Applications of Spanning Trees in K s,t"

Hannah Powell
6 years ago
Views:

1 Some Application of Spanning Tree in K,t L.H. Clark, A.T. Mohr, and T.D. Porter Department of Mathematic Southern Illinoi Univerity Carbondale, IL Abtract We partition the et of panning tree contained in the complete graph K n into panning tree contained in the complete bipartite graph K,t. Thi claification how that ome propertie of panning tree in K n can be derived from tree in K,t. We ue Abel binomial theorem and the formula for panning tree in K,t to obtain a proof of Cayley theorem uing partial derivative. Some reult concerning non-iomorphic panning tree are preented. In particular we count thee tree for Q 3 and the Peteren graph. Keyword. Abel binomial theorem, Cayley theorem, hypercube, Peteren graph, panning tree 1 Introduction We ue the tandard notation and terminology which can be found, e.g., in [12]. Let τ(g) denote the number of labelled panning tree in a graph G. Let K n denote the complete graph of n vertice and K,t the complete bipartite graph with partite et containing and t vertice, repectively. It i well known, a in e.g. [2, 3, 4, 5, 6, 10] that τ(k n )=n n 2, n 2 (1) τ(k,t )= t 1 t 1,,t 1. (2) We remark that (1) i often referred to a Cayley theorem. Let + t = n, where 1 t. We have the following obervation: Theorem 1. With n 2, any panning tree T in K n i a panning tree in K,t for a unique pair (, t), with1 t and + t = n. Contact author. 1

2 Proof. Conider a panning tree T in K n. Becaue T i a connected bipartite graph it i uniquely 2-colorable. So contruct thi unique bipartition by properly 2-coloring the vertex et of T with color red (R) and blue (B). Let the number of red vertice be and the number of blue vertice be t; w.l.o.g., let t. We then have T i a panning tree in thi K,t. The convere i traightforward. Theorem 2. With + t = n, any panning tree in K,t i a panning tree in K n. Proof. Thi follow ince K,t i a panning ubgraph of K n. Theorem 3. n 1 τ(k,n )=2τ(K n ). =1 Proof. By combining Theorem 1 and 2 we ee that to find τ(k n )wecanenumerate all labelled panning tree in the poible K,t graph. The double count occur from the 2-coloring of the partite et. We now proceed to how the LHS of Theorem 3 implie the RHS yielding a calculu baed proof of Cayley theorem. We hall apply Abel binomial formula, [1], which tate that for any x, y, andz that: (x + y) n = x(x z) 1 (y + z) n. (3) =0 Theorem 4. τ(k,t ) = τ(k n ). In word, the formula for τ(k n ) can be derived from the formula for τ(k,t ). Proof. From (3) we have n(x + y) n 1 = x (x + y)n = and conequently, =0 (x z)(x z) 2 (y + z) n (4) 2 = n(n 1)(x + y)n 2 y x = (x z)(x z) 2 (n )(y + z) n 1. =0 (5) We alo have, n(x + y) n 1 = y (x + y)n = =0 x(x z) 1 (n )(y + z) n 1. (6) 2

3 By ubtituting x = n, y = 0, andz = 1 into (5) and (6), we obtain, repectively, (7) and (8): n n 1 = n (n ) 1 (7) Adding (7) to (8) give n n 1 = 2n n 1 = =1 =1 =1 ) (n ) n 1. (8) ( n n 1 (n ) 1 n, (9) which yield the equation in Theorem 3. Thi give a proof of Cayley theorem uing partial derivative. The identity in (9) can alo be found in [2, 8, 11]. The idea in Theorem 1 and 2 are alo valid when graph are unlabelled, ince the unique bipartition apect i a tructural property of the graph T. So, for a connected graph G, let I(G) be the number of non-iomorphic panning tree in G. We have: Theorem 5. I(K n )= n/2 =1 I(K,n ). AformulaforI(K,t ) would then give a formula for I(K n ). We wrote a computer program that generate the et of labelled panning tree in a graph G. It then partition thi et into it iomorphim clae. Table 1 give ome reult found when G = K,t, I(K 5,5 ) being the larget calculation in term of computing time we have been able to produce. The top number in row and column t correpond to τ(k,t ) and the bottom number i I(K,t ). We have not een thee number in Table 1 in the literature. Obervational example of Theorem 5 and Table 1, uing well known value of ome I(K n ), are: and I(K 6 )=6=I(K 1,5 )+I(K 2,4 )+I(K 3,3 ) =1+2+3, I(K 7 )=11=I(K 1,6 )+I(K 2,3 )+I(K 3,4 ) =1+3+7, I(K 10 ) = 106 = I(K 1,9 )+I(K 2,8 )+I(K 3,7 )+I(K 4,6 )+I(K 5,5 ) = We would like to derive a general or aymptotic formula for I(K,t ). An aymptotic formula for I(K n ) i given by Otter [9] I(K n ) pn 5/2 r n, where p and r are contant. 3

4 n Table 1: Value of τ and I for K,t Our work o far ha given partition number for I(K 2,n )andi(k 3,n ). Let p k (n) denote the number of partition of an integer n into k or fewer part. Then, we have: n 1 I(K 2,n )=p 2 (n 1) = +1, for n 2 (10) 2 n 2 I(K 3,n )=p 3 (n 1) + p 2 (n 2 k), for n 4. (11) k=0 In (11), we adopt the convention that p 2 (0) = 1. A an example of (11), I(K 3,5 )=10=p 3 (4) + p 2 (3) + p 2 (2) + p 2 (1) + p 2 (0) = We ran our tree iomorphim program on ome other popular graph, namely Q 3 and the Peteren graph. Let Q n denote the n-dimenional cube and let P denote the Peteren graph. τ(q n ) i known, the value τ(q 3 ) = 384 and τ(p ) = 2000 are generally known, however, it appear that I(Q n )andi(p) may not be o univerally known. After applying our algorithm to Q 3 and P, we have found that I(Q 3 )=6andI(P) = 20. Table 2 give the breakdown of the ize of each iomorphim cla in Q 3. For example, row 2 denote that there are 3 clae, each containing 48 tree. Table 3 give drawing of a repreentative tree from each of the 6-clae. We remark the cla containing the panning path ha 72 tree. Table 4 give the different cla ize for the Peteren graph. On Autin Mohr webite [7], there are drawing of repreentative tree for the Peteren graph imilar to Table 3. There are alo drawing for the tree given in Table 1. 4

5 T = 384 I =6 Ditribution of Cla Size Num Tree Size of Cla Total 384 Table 2: Q 3 5

6 T = 384 I =6 48 in cla 48 in cla 144 in cla 48 in cla 72 in cla 24 in cla Table 3: Q 3 6

7 T = 2000 I =20 Ditribution of Cla Size Num Tree Size of Cla Total Table 4: Peteren Graph 7

8 Reference [1] N.H. Abel, Bewei eine Audruck, von welchem die Bionimial-Formel ein einzelner Fall it. J. reine angew. Math. 1, (1826), Reprinted in Œuvre Complète, 2nd ed., 1, (1881), [2] T.L. Autin, The enumeration of point labelled chromatic graph and tree, Canad. J. Math. 12, (1960), [3] L. Clark, On the enumeration of multipartite panning tree of the complete graph, Bull. of the ICA 38 (2003), [4] O. Eǧecioǧlu and J.B. Remmel, A bijection for panning tree of complete multipartite graph, Congreu Numerantium 100 (1994), [5] I.J. Good, The generalization of Lagrange expanion and the enumeration of tree, Proc. of the Cambridge Phil. Soc. 61 (1965), [6] R.P. Lewi, The number of panning tree of a complete multipartite graph, Dicrete Math. 197/198 (1999), [7] A. Mohr, amohr/. [8] J.W. Moon, Counting labeled tree, Canad. Math. Monograph No. 1, Canad. Math. Congre, [9] R. Otter, The number of tree, Ann. of Math. 49 (1948), [10] T.D. Porter, Generating the lit of panning tree in K,t, J. of Comb. Math. and Comb. Comput. 50 (2004), [11] L. Székely, Abel binomial Theorem, zekely/abel.pdf. [12] D.B. Wet, Introduction to Graph Theory, Second Edition, Prentice-Hall, Upper Saddle River, NJ,

List coloring hypergraphs

List coloring hypergraphs Lit coloring hypergraph Penny Haxell Jacque Vertraete Department of Combinatoric and Optimization Univerity of Waterloo Waterloo, Ontario, Canada pehaxell@uwaterloo.ca Department of Mathematic Univerity