22 August, 22
Graph decomposition Let G and H be graphs on m n vertices. A decompostion of G into copies of H is a collection {H i } of subgraphs of G such that each H i = H, and every edge of G belongs to exactly one H i. When G is complete, these objects are also known as H-designs.
Graph decomposition Let G and H be graphs on m n vertices. A decompostion of G into copies of H is a collection {H i } of subgraphs of G such that each H i = H, and every edge of G belongs to exactly one H i. When G is complete, these objects are also known as H-designs.
Example. Let G and H be the graphs shown below. G H
Example. Let G and H be the graphs shown below. G H
decomposition Relaxation: a fractional decomposition of G into H is a collection {(H i, w i )}, where w i is a positive real (w.l.o.g. rational) weight assigned to subgraph H i = H of G such that for every edge e of G, i : H i has e w i = 1. If there exists a fractional decomposition of G into H, let s write H Q G.
Example. Let P 2 be the path on two edges. Then P 2 Q C 5 by taking each of the 5 embeddings of P 2 in C 5 with weight 1 2. Example. Let H have n vertices. It is clear that H Q K n : simply take all copies of H on the points of G with suitable (constant) weight. By contrast, H Q G may sometimes fail; for instance, H G is necessary. In fact, we need the existsence of many subgraphs of H in G.
Example. Let P 2 be the path on two edges. Then P 2 Q C 5 by taking each of the 5 embeddings of P 2 in C 5 with weight 1 2. Example. Let H have n vertices. It is clear that H Q K n : simply take all copies of H on the points of G with suitable (constant) weight. By contrast, H Q G may sometimes fail; for instance, H G is necessary. In fact, we need the existsence of many subgraphs of H in G.
Example. Let P 2 be the path on two edges. Then P 2 Q C 5 by taking each of the 5 embeddings of P 2 in C 5 with weight 1 2. Example. Let H have n vertices. It is clear that H Q K n : simply take all copies of H on the points of G with suitable (constant) weight. By contrast, H Q G may sometimes fail; for instance, H G is necessary. In fact, we need the existsence of many subgraphs of H in G.
Non-Example. C 5 obviously does not admit a fractional decomposition into K 3, even though there is a signed weighting of triangles on 5 points which yields C 5 : 00 11 00 11
Non-Example. C 5 obviously does not admit a fractional decomposition into K 3, even though there is a signed weighting of triangles on 5 points which yields C 5 : 00 11 00 11
Non-Example. C 5 obviously does not admit a fractional decomposition into K 3, even though there is a signed weighting of triangles on 5 points which yields C 5 : 00 11 00 11
Non-Example. C 5 obviously does not admit a fractional decomposition into K 3, even though there is a signed weighting of triangles on 5 points which yields C 5 : 00 11 00 11
Non-Example. C 5 obviously does not admit a fractional decomposition into K 3, even though there is a signed weighting of triangles on 5 points which yields C 5 : 00 11 00 11
Adjacency matrices Let G, H have adjacency matrices A G, A H, respectively. Then G Q H is equivalent to a decomposition A G = i w i Q i A H Q i, where each Q i is an m n (0, 1)-matrix having exactly one 1 per column, and the w i are positive rationals.
We can extend all these ideas to t-uniform hypergraphs, or t-graphs for short. In what follows, think of a fixed, small t-graph H. Question. How dense (say in terms of (t 1)-subset degrees) does a large t-graph G need to be in order to guarantee H Q G? By transitivity of Q, it is enough to answer this question for complete t-graphs H = K t k. But this differs from a t-design in two respects. The decompositions here are fractional. Only a dense collection of prescribed edges is to be covered.
We can extend all these ideas to t-uniform hypergraphs, or t-graphs for short. In what follows, think of a fixed, small t-graph H. Question. How dense (say in terms of (t 1)-subset degrees) does a large t-graph G need to be in order to guarantee H Q G? By transitivity of Q, it is enough to answer this question for complete t-graphs H = K t k. But this differs from a t-design in two respects. The decompositions here are fractional. Only a dense collection of prescribed edges is to be covered.
We can extend all these ideas to t-uniform hypergraphs, or t-graphs for short. In what follows, think of a fixed, small t-graph H. Question. How dense (say in terms of (t 1)-subset degrees) does a large t-graph G need to be in order to guarantee H Q G? By transitivity of Q, it is enough to answer this question for complete t-graphs H = K t k. But this differs from a t-design in two respects. The decompositions here are fractional. Only a dense collection of prescribed edges is to be covered.
Main Theorem. For integers k t 2, there exists v 0 (t, k) and C = C(t) such that, for v > v 0 and ɛ < Ck 2t, any (1 ɛ)-dense t-graph G on v vertices admits a rational decomposition into copies of K t k. Proof idea. Estimate perturbations of the Johnson scheme s eigenvalues. From this, show that a certain square linear system has nonnegative solutions for small ɛ, by Cramer s rule. Interpret such solutions as fractional decompositions.
Main Theorem. For integers k t 2, there exists v 0 (t, k) and C = C(t) such that, for v > v 0 and ɛ < Ck 2t, any (1 ɛ)-dense t-graph G on v vertices admits a rational decomposition into copies of K t k. Proof idea. Estimate perturbations of the Johnson scheme s eigenvalues. From this, show that a certain square linear system has nonnegative solutions for small ɛ, by Cramer s rule. Interpret such solutions as fractional decompositions.
Set up Let V be a v-set and put X = ( V k ). For a set system F X, write F(T ) for the number of times T is covered by F. For fixed U ( ) V ( t, consider the family F = X [U] of all v t k t) k-subsets of V which contain U. Then ( ) v T U X [U](T ) =, k T U since this counts the number of k-subsets containing both T and U.
Set up Let V be a v-set and put X = ( V k ). For a set system F X, write F(T ) for the number of times T is covered by F. For fixed U ( ) V ( t, consider the family F = X [U] of all v t k t) k-subsets of V which contain U. Then ( ) v T U X [U](T ) =, k T U since this counts the number of k-subsets containing both T and U.
Set up Let V be a v-set and put X = ( V k ). For a set system F X, write F(T ) for the number of times T is covered by F. For fixed U ( ) V ( t, consider the family F = X [U] of all v t k t) k-subsets of V which contain U. Then ( ) v T U X [U](T ) =, k T U since this counts the number of k-subsets containing both T and U.
Therefore, where ξ i = for i = 0, 1,..., t. X [U](T ) = ξ T \U, ( ) v t i = v k t i k t i (k t i)! + o(v k t i ). Let n = ( v t). Define the n n matrix M by for T, U ( V t ). M(T, U) = ξ T \U = X [U](T ),
Therefore, where ξ i = for i = 0, 1,..., t. X [U](T ) = ξ T \U, ( ) v t i = v k t i k t i (k t i)! + o(v k t i ). Let n = ( v t). Define the n n matrix M by for T, U ( V t ). M(T, U) = ξ T \U = X [U](T ),
In fact, M factors as M = WW, where W is the well-known inclusion matrix of t-subsets versus k-subsets. Note that M = M and the constant column (row) sum of M is ( )( ) v t k. k t t
In fact, M factors as M = WW, where W is the well-known inclusion matrix of t-subsets versus k-subsets. Note that M = M and the constant column (row) sum of M is ( )( ) v t k. k t t
Note that a nonnegative solution x to Mx = 1 induces a fractional decomposition K t k Q K t v. Simply take each X [U] with weight x(u), and the total coverage at T is x(u)m(t, U) = (Mx)(T ) = 1 U on each t-set T.
Restriction to edges of G But decomposing a non-complete t-graph G is not so easy. We must restrict our attention to k-subsets that cover only those edges present in G. To this end, define X G as the family of all k-subsets which induce a clique in G. In other words, K X G if and only if K V with K = k, and T K with T = t implies T is an edge of G. Now consider X G [U], the family of all k-subsets on V which contain U and also induce a clique in G. Define the G G matrix M, with rows and columns indexed by edges of G, by M(T, U) = X G [U](T ).
Restriction to edges of G But decomposing a non-complete t-graph G is not so easy. We must restrict our attention to k-subsets that cover only those edges present in G. To this end, define X G as the family of all k-subsets which induce a clique in G. In other words, K X G if and only if K V with K = k, and T K with T = t implies T is an edge of G. Now consider X G [U], the family of all k-subsets on V which contain U and also induce a clique in G. Define the G G matrix M, with rows and columns indexed by edges of G, by M(T, U) = X G [U](T ).
Restriction to edges of G But decomposing a non-complete t-graph G is not so easy. We must restrict our attention to k-subsets that cover only those edges present in G. To this end, define X G as the family of all k-subsets which induce a clique in G. In other words, K X G if and only if K V with K = k, and T K with T = t implies T is an edge of G. Now consider X G [U], the family of all k-subsets on V which contain U and also induce a clique in G. Define the G G matrix M, with rows and columns indexed by edges of G, by M(T, U) = X G [U](T ).
Note M is symmetric, and it is approximately a principal submatrix of M in the positions indexed by G. And, most importantly, a nonnegative solution x to Mx = 1 if it exists, yields a fractional decomposition K t k Q G.
Counting the missing cliques Lemma. Suppose G is a (1 ɛ)-dense simple t-graph. 1. Given an edge T and i with 0 i t, there are at least ( )( ) [ ( ) ] t v t + i 1 ɛ + o(1) i i i edges U such that T \ U = i and T U induces a clique in G. 2. If T and U are edges of G with T \ U = i and such that T U induces a clique in G, then there are at least ( ] t + i ( v t i k t i ) [ 1 (( k t ) i )) ɛ + o(1) k-subsets containing T U and inducing a clique in G. Proof: easy counting Remarks: Part (a) essentially asserts that most entries of M are nonzero, while part (b) asserts that those nonzero entries are close to those of M.
Counting the missing cliques Lemma. Suppose G is a (1 ɛ)-dense simple t-graph. 1. Given an edge T and i with 0 i t, there are at least ( )( ) [ ( ) ] t v t + i 1 ɛ + o(1) i i i edges U such that T \ U = i and T U induces a clique in G. 2. If T and U are edges of G with T \ U = i and such that T U induces a clique in G, then there are at least ( ] t + i ( v t i k t i ) [ 1 (( k t ) i )) ɛ + o(1) k-subsets containing T U and inducing a clique in G. Proof: easy counting Remarks: Part (a) essentially asserts that most entries of M are nonzero, while part (b) asserts that those nonzero entries are close to those of M.
Eigenvalues The eigenvalues of M are easy to get explicitly from Delsarte theory. Since M = t i=0 ξ ia i, it follows that M = t j=0 θ je j, where θ j = t ξ i P ij i=0... = ( )( ) k j v t + o(v k t ). t j k t Recall that the E i are orthogonal idempotents. So these coefficients θ i are the eigenvalues of M.
Perturbation Lemma. Suppose A and A are Hermitian matrices such that every eigenvalue of A is greater than A. Then A + A is positive definite. Let s invoke this with A = M G and A = M := M M G. We know that every eigenvalue of A is at least θ t The earlier lemma offers an estimate A < θ t ( k t ) 2 ɛ + o(v k t ).
Cramer s rule Finally, for Cramer s rule, we need to estimate eigenvalues of the matrix obtained by replacing a column of M by 1. This is accomplished by interlacing results, which guarantee a lower bound of 1 2 θ t. Conclusion: Our solution vector M 1 1 is (asymptotically in v) entrywise positive for ɛ < 1 2 ( ) k 2. t
Cramer s rule Finally, for Cramer s rule, we need to estimate eigenvalues of the matrix obtained by replacing a column of M by 1. This is accomplished by interlacing results, which guarantee a lower bound of 1 2 θ t. Conclusion: Our solution vector M 1 1 is (asymptotically in v) entrywise positive for ɛ < 1 2 ( ) k 2. t
An application decomposition of certain high-degree circulants has proved useful in balanced sampling plans. Here, one wishes to fairly sample, by k-subsets, all pairs of elements which are not too close together. The rational weights w i associated with each sample H i = Kk is the probability with which that sample is selected for testing. The total probability with which a pair of points is tested is either zero or one, according to the definition of close. So fractional decompositions are useful in statistics.
P. Dukes, Rational decomposition of dense hypergraphs and some related eigenvalue estimates, LAA, 22. P. Dukes and A.C.H. Ling, Existence of balanced sampling plans avoiding cyclic distances, Metrika, 2007. C.D. Godsil, Notes on Association Schemes. University of Waterloo, 2005. R. Yuster, decompositions of dense hypergraphs. Approximation, randomization and combinatorial optimization, 2005. - THE END -