On the Maximum Number of Cliques in a Graph

A clique is a set of pairwise adjacent vertices in a graph. We determine the maximum number of cliques in a graph for the following graph classes: (1) graphs with n vertices and m edges; (2) graphs with n vertices, m edges, and maximum degree Δ; (3) d-degenerate graphs with n vertices and m edges; (4) planar graphs with n vertices and m edges; and (5) graphs with n vertices and no K₅-minor or no K_3,3-minor. For example, the maximum number of cliques in a planar graph with n vertices is 8(n − 2). Research supported by a Marie Curie Fellowship of the European Community under contract 023865, and by the projects MCYT-FEDER BFM2003-00368 and Gen. Cat 2001SGR00224.     

On embeddability and stresses of graphs

Gluck has proven that triangulated 2-spheres are generically 3-rigid. Equivalently, planar graphs are generically 3-stress free. We show that already the K ₅-minor freeness guarantees the stress freeness. More generally, we prove that every K _r+2-minor free graph is generically r-stress free for 1≤r≤4. (This assertion is false for r≥6.) Some further extensions are discussed. Supported by an I.S.F. grant.     

A Note on Group Choosability of Graphs with Girth at least 4

In this paper we show that the group choice number of a graph without K ₅-minor or K _3,3-minor with girth at least 4 (resp. 6) is at most 4 (resp. 3) and we conclude that these results hold for the group chromatic number, the choice number and the chromatic number.     

Distance-transitive dihedrants

The main result of this article is a classification of distance-transitive Cayley graphs on dihedral groups. We show that a Cayley graph X on a dihedral group is distance-transitive if and only if X is isomorphic to one of the following graphs: the complete graph K _2n ; a complete multipartite graph K _t×m with t anticliques of size m, where t m is even; the complete bipartite graph without 1-factor K _n,n − nK ₂; the cycle C _2n ; the incidence or the non-incidence graph of the projective geometry PG _d-1(d,q), d ≥ 2; the incidence or the non-incidence graph of a symmetric design on 11 vertices.     

On K 1,k -factorization of bipartite multigraphs

A K1,k-factorization of λKm,n is a set of edge-disjoint K1,k-factors of λKm,n, which partition the set of edges of λKm,n. In this paper, it is proved that a sufficient condition for the existence of K1,k-factorization of λKm,n, whenever k is any positive integer, is that （1） m ≤ kn, （2） n ≤ km, （3） km-n = kn-m ≡ 0 （mod （k^2- 1）） and （4） λ（km-n）（kn-m） ≡ 0 （mod k（k- 1）（k^2 - 1）（m ＋ n））.     

P4k?1-factorization of bipartite multigraphs

Let λK _m,n be a bipartite multigraph with two partite sets having m and n vertices, respectively. A P _v-factorization of λK _m,n is a set of edge-disjoint P _v -factors of λK _m,n which partition the set of edges of λK _m,n. When v is an even number, Ushio, Wang and the second author of the paper gave a necessary and sufficient condition for the existence of a P _v -factorization of λK _m,n. When v is an odd number, we proposed a conjecture. However, up to now we only know that the conjecture is true for v = 3. In this paper we will show that the conjecture is true when v = 4k − 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P _4k−1-factorization of λK _m,n is (1) (2k − 1)m ⩽ 2kn, (2) (2k − 1)n ⩽ 2km, (3) m + n ≡ 0 (mod 4k − 1), (4) λ(4k − 1)mn/[2(2k − 1)(m + n)] is an integer.     

Topological minors in bipartite graphs

For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and for integers s and t such that 2 ≤ s ≤ t, 0 ≤ m − s ≤ n − t, and m + n ≤ 2s + t − 1, we prove that if G has at least mn − (2(m − s) + n − t) edges then it contains a subdivision of the complete bipartite K _(s,t) with s vertices in the m-class and t vertices in the n-class. Furthermore, we characterize the corresponding extremal bipartite graphs with mn − (2(m − s) + n − t + 1) edges for this topological Turan type problem.     

On Some Three-Color Ramsey Numbers

 In this paper we study three-color Ramsey numbers. Let K _i,j denote a complete i by j bipartite graph. We shall show that (i) for any connected graphs G ₁, G ₂ and G ₃, if r(G ₁, G ₂)≥s(G ₃), then r(G ₁, G ₂, G ₃)≥(r(G ₁, G ₂)−1)(χ(G ₃)−1)+s(G ₃), where s(G ₃) is the chromatic surplus of G ₃; (ii) (k+m−2)(n−1)+1≤r(K _1,k , K _1,m , K _n )≤ (k+m−1)(n−1)+1, and if k or m is odd, the second inequality becomes an equality; (iii) for any fixed m≥k≥2, there is a constant c such that r(K _k,m , K _k,m , K _n )≤c(n/logn), and r(C _2m , C _2m , K _n )≤c(n/logn) ^m/(m−1) for sufficiently large n. Received: July 25, 2000 Final version received: July 30, 2002 RID="*" ID="*" Partially supported by RGC, Hong Kong; FRG, Hong Kong Baptist University; and by NSFC, the scientific foundations of education ministry of China, and the foundations of Jiangsu Province Acknowledgments. The authors are grateful to the referee for his valuable comments. AMS 2000 MSC: 05C55     

Potentially <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> — <Emphasis Type="Italic">G</Emphasis>-graphical sequences: A survey

The set of all non-increasing nonnegative integer sequences π = (d(v ₁), d(v ₂), …, d(v _n )) is denoted by NS _n . A sequence π ∈ NS _n is said to be graphic if it is the degree sequence of a simple graph G on n vertices, and such a graph G is called a realization of π. The set of all graphic sequences in NS _n is denoted by GS _n . A graphical sequence π is potentially H-graphical if there is a realization of π containing H as a subgraph, while π is forcibly H-graphical if every realization of π contains H as a subgraph. Let K _k denote a complete graph on k vertices. Let K _m −H be the graph obtained from Km by removing the edges set E(H) of the graph H (H is a subgraph of K _m ). This paper summarizes briefly some recent results on potentially K _m −G-graphic sequences and give a useful classification for determining σ (H, n).     

Graphic Sequences with a Realization Containing a Union of Cliques

An integer sequence π is said to be graphic if it is the degree sequence of some simple graph G. In this case we say that G is a realization of π. Given a graph H, and a graphic sequence π we say that π is potentially H-graphic if there is some realization of π that contains H as a subgraph. We define σ(H,n) to be the minimum even integer such that every graphic sequence with sum at least σ(H,n) is potentially H-graphic. In this paper, we determine σ(H,n) for the graph H = K_m1∪ K_m2∪...∪ K_mk when n is a sufficiently large integer. This is accomplished by determining σ(K_j + kK₂,n) where j and k are arbitrary positive integers, and considering the case where j = m − 2k and m = ∑ m_i.     

Minimum Resolvable Coverings of K v with Copies of K 4 ? e

Suppose K _v is the complete undirected graph with v vertices and K ₄ − e is the graph obtained from a complete graph K ₄ by removing one edge. Let (K ₄ − e)-MRC(v) denote a resolvable covering of K _v with copies of K ₄ − e with the minimum possible number n(v, K ₄ − e) of parallel classes. It is readily verified that n(v, K₄-e) 3 é2(v-1)/5 ù{n(v, K_4-e) \geq \lceil 2(v-1)/5 \rceil} . In this article, it is proved that there exists a (K ₄ − e)-MRC(v) with é2(v-1)/5 ù{\lceil 2(v-1)/5 \rceil} parallel classes if and only if v ≡ 0 (mod 4) with the possible exceptions of v = 108, 172, 228, 292, 296, 308, 412. In addition, the known results on the existence of maximum resolvable (K ₄ − e)-packings are also improved.     

The existence of a 2‐factor in K1, n‐free graphs with large connectivity and large edge‐connectivity

In this article, we study the existence of a 2‐factor in a K_{1, n}‐free graph. Sumner [J London Math Soc 13 (1976), 351–359] proved that for n?4, an (n?1)‐connected K_{1, n}‐free graph of even order has a 1‐factor. On the other hand, for every pair of integers m and n with m?n?4, there exist infinitely many (n?2)‐connected K_{1, n}‐free graphs of even order and minimum degree at least m which have no 1‐factor. This implies that the connectivity condition of Sumner's result is sharp, and we cannot guarantee the existence of a 1‐factor by imposing a large minimum degree. On the other hand, Ota and Tokuda [J Graph Theory 22 (1996), 59–64] proved that for n?3, every K_{1, n}‐free graph of minimum degree at least 2n?2 has a 2‐factor, regardless of its connectivity. They also gave examples showing that their minimum degree condition is sharp. But all of them have bridges. These suggest that the effects of connectivity, edge‐connectivity and minimum degree to the existence of a 2‐factor in a K_{1, n}‐free graph are more complicated than those to the existence of a 1‐factor. In this article, we clarify these effects by giving sharp minimum degree conditions for a K_{1, n}‐free graph with a given connectivity or edge‐connectivity to have a 2‐factor. Copyright © 2010 Wiley Periodicals, Inc. J Graph Theory 68:77‐89, 2011     

The proof of Ushio’s conjecture concerning path factorization of complete bipartite graphs

Let K _m,n be a complete bipartite graph with two partite sets having m and n vertices, respectively. A P _v -factorization of K _m,n is a set of edge-disjoint P _v -factors of K _m,n which partition the set of edges of K _m,n . When v is an even number, Wang and Ushio gave a necessary and sufficient condition for existence of P _v -factorization of K _m,n . When k is an odd number, Ushio in 1993 proposed a conjecture. Very recently, we have proved that Ushio’s conjecture is true when v = 4k − 1. In this paper we shall show that Ushio Conjecture is true when v = 4k − 1, and then Ushio’s conjecture is true. That is, we will prove that a necessary and sufficient condition for the existence of a P _4k+1-factorization of K _m,n is (i) 2km≤(2k+1)n, (ii) 2kn≤(2k+1)m, (iii) m+n≡0 (mod 4k+1), (iv) (4k+1)mn/[4k(m+n)] is an integer.     

The Smallest Degree Sum That Yields Potentially Kr＋1 - K3-Graphic Sequences

Let a（Kr,＋1 - K3,n） be the smallest even integer such that each n-term graphic sequence п= （d1,d2,…dn） with term sum σ（п） = d1 ＋ d2 ＋…＋ dn 〉 σ（Kr＋1 -K3,n） has a realization containing Kr＋1 - K3 as a subgraph, where Kr＋1 -K3 is a graph obtained from a complete graph Kr＋1 by deleting three edges which form a triangle. In this paper, we determine the value σ（Kr＋1 - K3,n） for r ≥ 3 and n ≥ 3r＋ 5.     

New Lower Bounds on the Multicolor Ramsey Numbers R r (C 2 m )

The multicolor Ramsey number R_r(H) is defined to be the smallest integer n=n(r) with the property that any r-coloring of the edges of the complete graph K_n must result in a monochromatic subgraph of K_n isomorphic to H. It is well known that 2^rm<R_r(C_2m+1)<2(r+2)!m and R_r(C_2m)≥(r−1)(m−1)+1. In this paper, we prove that R_r(C_2m)≥2(r−1)(m−1)+2. This research is supported by NSFC(60373096, 60573022) and SRFDP(20030141003)     

Edge Colouring of K2n with Spanning Star-Forests Receiving Distinct Colours

 Let S _ni be a star of size n _i and let S=S _n1∪…∪S _nk≠S _2n−3∪S ₁ or S ₂∪S ₂ be a spanning star-forest of the complete graph K _2n. We prove that K _2n has a proper (2n−1)-edge-colouring such that all the edges of S receive distinct colours. This result is very useful in the study of total-colourings of graphs. Received: March 8, 1995 / Revised: May 16, 1997     

On linguistic dynamical systems,families of graphs of large girth,and cryptography

The paper is devoted to the study of a linguistic dynamical system of dimension n ≥ 2 over an arbitrary commutative ring K, i.e., a family F of nonlinear polynomial maps f _α : K ⁿ → K ⁿ depending on “time” α ∈ {K − 0} such that f _α ⁻¹ = f _−αM, the relation f _α1 (x) = f _α2 (x) for some x ∈ Kⁿ implies α₁ = α₂, and each map f _α has no invariant points. The neighborhood {f _α (υ)∣α ∈ K − {0}} of an element v determines the graph Γ(F) of the dynamical system on the vertex set Kⁿ. We refer to F as a linguistic dynamical system of rank d ≥ 1 if for each string a = (α₁, υ, α₂), s ≤ d, where α_i + α_i+1 is a nonzero divisor for i = 1, υ, d − 1, the vertices υ _a = f _α1 × ⋯ × f _αs (υ) in the graph are connected by a unique path. For each commutative ring K and each even integer n ≠= 0 mod 3, there is a family of linguistic dynamical systems L_n(K) of rank d ≥ 1/3n. Let L(n, K) be the graph of the dynamical system L_n(q). If K = F_q, the graphs L(n, F_q) form a new family of graphs of large girth. The projective limit L(K) of L(n, K), n → ∞, is well defined for each commutative ring K; in the case of an integral domain K, the graph L(K) is a forest. If K has zero divisors, then the girth of K drops to 4. We introduce some other families of graphs of large girth related to the dynamical systems L_n(q) in the case of even q. The dynamical systems and related graphs can be used for the development of symmetric or asymmetric cryptographic algorithms. These graphs allow us to establish the best known upper bounds on the minimal order of regular graphs without cycles of length 4n, with odd n ≥ 3. Bibliography: 42 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 214–234.     

The number of K m,m -free graphs

A graph is called H-free if it contains no copy of H. Denote by f _n (H) the number of (labeled) H-free graphs on n vertices. Erdős conjectured that f _n (H) ≤ 2^{(1+o(1))ex(n,H)}. This was first shown to be true for cliques; then, Erdős, Frankl, and R?dl proved it for all graphs H with χ(H)≥3. For most bipartite H, the question is still wide open, and even the correct order of magnitude of log₂ f _n (H) is not known. We prove that f _n (K _m,m ) ≤ 2 ^O (n ^2−1/m ) for every m, extending the result of Kleitman and Winston and answering a question of Erdős. This bound is asymptotically sharp for m∈{2,3}, and possibly for all other values of m, for which the order of ex(n,K _m,m ) is conjectured to be Θ(n ^2−1/m ). Our method also yields a bound on the number of K _m,m -free graphs with fixed order and size, extending the result of Füredi. Using this bound, we prove a relaxed version of a conjecture due to Haxell, Kohayakawa, and Łuczak and show that almost all K _3,3-free graphs of order n have more than 1/20·ex(n,K _3,3) edges.     

On Independent Cycles in a Bipartite Graph

 Let G=(V ₁,V ₂;E) be a bipartite graph with 2k≤m=|V ₁|≤|V ₂|=n, where k is a positive integer. We show that if the number of edges of G is at least (2k−1)(n−1)+m, then G contains k vertex-disjoint cycles, unless e(G)=(2k−1)(n−1)+m and G belongs to a known class of graphs. Received: December 9, 1998 Final version received: June 2, 1999     

Graphs of Non-Crossing Perfect Matchings

 Let P _n be a set of n=2m points that are the vertices of a convex polygon, and let ℳ _m be the graph having as vertices all the perfect matchings in the point set P _n whose edges are straight line segments and do not cross, and edges joining two perfect matchings M ₁ and M ₂ if M ₂=M ₁−(a,b)−(c,d)+(a,d)+(b,c) for some points a,b,c,d of P _n . We prove the following results about ℳ _m : its diameter is m−1; it is bipartite for every m; the connectivity is equal to m−1; it has no Hamilton path for m odd, m>3; and finally it has a Hamilton cycle for every m even, m≥4. Received: October 10, 2000 Final version received: January 17, 2002 RID="*" ID="*" Partially supported by Proyecto DGES-MEC-PB98-0933 Acknowledgments. We are grateful to the referees for comments that helped to improve the presentation of the paper.