Long Snakes in Powers of the Complete Graph with an Odd Number of Vertices

In [5] Abbott and Katchalski ask if there exists a constantc < 0 such that for every d 2 there is a snake (cycle withoutchords) of length at least c3^d in the product of d copies ofthe complete graph K₃. We show that the answer to the abovequestion is positive, and that in general for any odd integern there is a constant c_n such that for every d 2 there is asnake of length at least c_n n^d in the product of d copies ofthe complete graph K_n.     

On the Decomposition of Graphs into Complete Bipartite Graphs

In a complete bipartite decomposition π of a graph, we consider the number ϑ(v;π) of complete bipartite subgraphs incident with a vertex v. Let ϑ(G)= ϑ(v;π). In this paper the exact values of ϑ(G) for complete graphs and hypercubes and a sharp upper bound on ϑ(G) for planar graphs are provided, respectively. An open problem proposed by P.C. Fishburn and P.L. Hammer is solved as well.     

An Upper Bound on the Basis Number of the Powers of the Complete Graphs

The basis number of a graph G is defined by Schmeichel to be the least integer h such that G has an h-fold basis for its cycle space. MacLane showed that a graph is planar if and only if its basis number is . Schmeichel proved that the basis number of the complete graph K _n is at most 3>. We generalize the result of Schmeichel by showing that the basis number of the d-th power of K _n is at most 2d+1.     

Decompositions of Complete Multipartite Graphs into Cycles of Even Length

Necessary and sufficient conditions for the existence of an edge-disjoint decomposition of any complete multipartite graph into even length cycles are investigated. Necessary conditions are listed and sufficiency is shown for the cases when the cycle length is 4, 6 or 8. Further results concerning sufficiency, provided certain “small” decompositions exist, are also given for arbitrary even cycle lengths. Revised: November 28, 1997     

Decomposing Complete Equipartite Graphs into Closed Trails of Length k

Necessary conditions for a simple connected graph G to admit a decomposition into closed trails of length k ≥ 3 are that G is even and its total number of edges is a multiple of k. In this paper we show that these conditions are sufficient in the case when G is the complete equipartite graph having at least three parts, each of the same size.     

关于完全t部图的色唯一性

设P(G,λ)是图的色多项式。如果对任意使P(G,λ)=P(H,λ)的图H都与G同构,则称图G是色唯一图.这里通过比较t 1色类的色划分数目,讨论了由Koh和Teo在文献[1]中提出的问题(若|ni-nj|≤2,当min(n1,n2,…,nt)充分大时,完全t部图K(n1,n2,…,nt)是否是色唯一图?)。改进了文献[5]中的结果。证明了若Σ1≤i≤ta2i=T,min{n a1,n a2,…,nt at,n-1}≥(T 1)/2,则K(n a1,n a2,…,n at)是色唯一图(其中ai是实数,n ai是正整数)。从而证明了若|ni-nj|≤k(i,j=1,2,…,t),min{n1,n2,…,nt}≥tk2/8 1,则K(n1,n2,…,nt)是色唯一图。     

On a Decomposition of Complete Graphs

It is shown, among other results, that for any prime power q, the complete graph on 1+q+q ²+q ³ vertices can be decomposed into a union of 1+q Siamese Strongly Regular Graphs S R G(1+q+q ²+q ³,q+q ²,q–1,q+1) sharing 1+q ² cliques of size 1+q. Acknowledgments.The authors are indebted to a referee for a very extensive report and for many suggestions which improved the presentation of the paper tremendously.AMS Subject Numbers: 05B05, 05B20, 05E30This work was completed while the first author was on sabbatical leave visiting Institute for studies in theoretical Physics and Mathematics, (IPM), in Tehran, Iran. Support and hospitality is appreciated. Supported by an NSERC operating grant.     

关于完全t部图K(n1,n2,…,nt)的色唯一性

设P（G，λ）是图G的色多项式，如果对任意使P（G，λ）=P（H，λ）的图H都与G同构，则称G是色唯一图。这里通过比较图的特征子图的个数，讨论了由Koh和Teo在文献[1]中提出的问题（若｜ni-nj｜≤2，1≤i，j≤t且min{n1，n2，…，nt}充分大，K（n1，n2，…，nt）是否为色唯一图？）。证明了，若｜ni—nj｜≤2且t↑∑↑i=1 ni〉t^2/2＋t√t-1，则K（n1，n2，…，nt）是色唯一图；若αi=0或k，t↑∑↑i=1 n＋αi〉t^2k^2/8＋｜tk｜/2√t-1，则K（n＋α1，n＋α2，…，n＋αt）是色唯一图。其条件比文献[4]中的条件较好一些。     

On the Number of Genus Embeddings of Complete Bipartite Graphs

In this paper, on the basis of joint tree model introduced by Liu, by dividing the associated surfaces into segments layer by layer, we show that there are at least ${C_{1}\cdot C_{2}^{\frac{m}{2}}\cdot C_{3}^{\frac{n}{2}}(m-1)^{m-\frac{1}{2}}(n-1)^{n-\frac{1}{2}}}$ distinct genus embeddings for complete bipartite graph K _m,n , where C ₁, C ₂, and C ₃ are constants depending on the residual class of m modular 4 and that of n modular 4.     

On Idomatic Partitions of Direct Products of Complete Graphs

Independent dominating sets in the direct product of four complete graphs are considered. Possible types of such sets are classified. The sets in which every pair of vertices agree in exactly one coordinate, called T ₁-sets, are explicitly described. It is proved that the direct product of four complete graphs admits an idomatic partition into T ₁-sets if and only if each factor has at least three vertices and the orders of at least two factors are divisible by 3.     

Families of Graphs Closed Under Taking Powers

 This paper gives simple proofs for “G ^k ∈? implies G ^{k +1}∈?” when ? is the family of all interval graphs, all proper interval graphs, all cocomparability graphs, or all m-trapezoid graphs. Received: November 21, 1997 Final version received: October 5, 1998     

Decompositions of Complete Graphs

If s₁, s₂, ..., s_t are integers such that n – 1 = s₁ +s₂ + ... + s_t and such that for each i (1 i t), 2 s_i n –1 and s_in is even, then K_n can be expressed as the union G₁G₂...G_tof t edge-disjoint factors, where for each i, G_i is s_i-regularand s_i-connected. Moreover, whenever s_i = s_j, G_i and G_j areisomorphic. 1991 Mathematics Subject Classification 05C70.     

Common Multiples of Complete Graphs

A graph H is said to divide a graph G if there exists a setS of subgraphs of G, all isomorphic to H, such that the edgeset of G is partitioned by the edge sets of the subgraphs inS. Thus, a graph G is a common multiple of two graphs if eachof the two graphs divides G. This paper considers common multiples of a complete graph oforder m and a complete graph of order n. The complete graphof order n is denoted K_n. In particular, for all positive integersn, the set of integers q for which there exists a common multipleof K₃ and K_n having precisely q edges is determined. It is shown that there exists a common multiple of K₃ and K_nhaving q edges if and only if q 0 (mod 3), q 0 (mod n2) and (1) q 3 n2 when n 5 (mod 6); (2) q (n + 1) n2 when n is even; (3) q {36, 42, 48} when n = 4. The proof of this result uses a variety of techniques includingthe use of Johnson graphs, Skolem and Langford sequences, andequitable partial Steiner triple systems. 2000 MathematicalSubject Classification: 05C70, 05B30, 05B07.     

Ramsey Numbers of Some Bipartite Graphs Versus Complete Graphs

The Ramsey number r(H, K _n ) is the smallest positive integer N such that every graph of order N contains either a copy of H or an independent set of size n. The Turán number ex(m, H) is the maximum number of edges in a graph of order m not containing a copy of H. We prove the following two results: (1) Let H be a graph obtained from a tree F of order t by adding a new vertex w and joining w to each vertex of F by a path of length k such that any two of these paths share only w. Then r(H,K_n) ￡ c_k,t [(n^1+1/k)/(ln^1/k n)]{r(H,K_n)\leq c_{k,t}\, {n^{1+1/k}\over \ln^{1/k} n}} , where c _k,t is a constant depending only on k and t. This generalizes some results in Li and Rousseau (J Graph Theory 23:413–420, 1996), Li and Zang (J Combin Optim 7:353–359, 2003), and Sudakov (Electron J Combin 9, N1, 4 pp, 2002). (2) Let H be a bipartite graph with ex(m, H) = O(m ^γ ), where 1 < γ < 2. Then r(H,K_n) ￡ c_H ([(n)/(lnn)])^1/(2-g){r(H,K_n)\leq c_H ({n\over \ln n})^{1/(2-\gamma)}}, where c _H is a constant depending only on H. This generalizes a result in Caro et al. (Discrete Math 220:51–56, 2000).     

Chaotic Numbers of Complete Bipartite Graphs and Tripartite Graphs

In this paper, we study the chaotic numbers of complete bipartite graphs and complete tripartite graphs. For the complete bipartite graphs, we find closed-form formulas of the chaotic numbers and characterize all chaotic mappings. For the complete tripartite graphs, we develop an algorithm running in O(n ⁴ ₃) time to find the chaotic numbers, with n ₃ the number of vertices in the largest partite set.Research supported by NSC 90-2115-M-036-003.The author thanks the authors of Ref. 6, since his work was motivated by their work. Also, the author thanks the referees for helpful comments which made the paper more readable.     

Disjoint Edges in Complete Topological Graphs

It is shown that every complete $n$ -vertex simple topological graph has at $\varOmega (n^{1/3})$ pairwise disjoint edges, and these edges can be found in polynomial time. This proves a conjecture of Pach and Tóth, which appears as Problem 5 from Chapter 9.5 in Research Problems in Discrete Geometry by Brass, Moser, and Pach.     

Unavoidable Configurations in Complete Topological Graphs

A topological graph is a graph drawn in the plane so that its vertices are represented by points, and its edges are represented by Jordan curves connecting the corresponding points, with the property that any two curves have at most one point in common. We define two canonical classes of topological complete graphs, and prove that every topological complete graph with n vertices has a canonical subgraph of size at least clog^1/8 n, which belongs to one of these classes. We also show that every complete topological graph with n vertices has a non-crossing subgraph isomorphic to any fixed tree with at most clog^1/6 n vertices.     

Seidel Integral Complete r-Partite Graphs

A graph is S-integral (or Seidel integral) if the spectrum of its Seidel matrix consists entirely of integers. In this paper, we give a sufficient and necessary condition for complete r-partite graphs to be S-integral, from which we construct infinitely many new classes of S-integral graphs. We also present an upper bound and a lower bound for the smallest S-eigenvalue (or Seidel eigenvalue) of a complete multipartite graph.     

Codes Based on Complete Graphs

We consider the problem of embedding the even graphical code based on the complete graph onn vertices into a shortening of a Hamming code of length 2^m-1, wherem = h(n) should be as small as possible. As it turns out, this problem is equivalent to the existence problem for optimal codes with minimum distance 5, and optimal embeddings can always be realized as graphical codes based onK _n. As a consequence, we are able to determineh(n) exactly for alln of the form 2 ^k + 1 and to narrow down the possibilities in general to two or three conceivable values.Dedicated to Hanfried Lenz on the occasion of his 80th birthdayThe research for this note was done while the first author was visiting the University of Waterloo and the University of Rome, respectively. He thanks his colleagues there for their hospitality and also acknowledges the financial support of the Consiglio Nazionale delle Ricerche (Italy). The third author acknowledges the support of the National Science and Engineering Research Council of Canada given under grant #0GP0009258.