A degree condition for the circumference of a graph

We present a new condition on the degree sums of a graph that implies the existence of a long cycle. Let c(G) denote the length of a longest cycle in the graph G and let m be any positive integer. Suppose G is a 2-connected graph with vertices x₁,…,x_n and edge set E that satisfies the property that, for any two integers j and k with j < k, x_jx_k ? E, d(x_i) ? j and d(x_k) ? K - 1, we have (1) d(x_i) + d(x_k ? m if j + k ? n and (2) if j + k < n, either m ? n or d(x_j) + d(x_k) ? min(K + 1,m). Then c(G) ? min(m, n). This result unifies previous results of J.C. Bermond and M. Las Vergnas, respectively.     

Mutually orthogonal graph squares

A decomposition ??={G₁, G₂,…,G_s} of a graph G is a partition of the edge set of G into edge‐disjoint subgraphs G₁, G₂,…,G_s. If G_i?H for all i∈{1, 2, …, s}, then ?? is a decomposition of G by H. Two decompositions ??={G₁, G₂, …, G_n} and ?={F₁, F₂,…,F_n} of the complete bipartite graph K_n,n are orthogonal if |E(G_i)∩E(F_j)|=1 for all i,j∈{1, 2, …, n}. A set of decompositions {??₁, ??₂, …, ??_k} of K_{n, n} is a set of k mutually orthogonal graph squares (MOGS) if ??_i and ??_j are orthogonal for all i, j∈{1, 2, …, k} and i≠j. For any bipartite graph G with n edges, N(n, G) denotes the maximum number k in a largest possible set {??₁, ??₂, …, ??_k} of MOGS of K_{n, n} by G. El‐Shanawany conjectured that if p is a prime number, then N(p, P_{p+ 1})=p, where P_{p+ 1} is the path on p+ 1 vertices. In this article, we prove this conjecture. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 369–373, 2009     

Optimal labelling of a product of two paths

If G is a graph with p vertices and at least one edge, we set φ (G) = m n max |f(u) ? f(v)|, where the maximum is taken over all edges uv and the minimum over all one-to-one mappings f : V(G) → {1, 2, …, p}: V(G) denotes the set of vertices of G.P_n will denote a path of length n whose vertices are integers 1, 2, …, n with i adjacent to j if and only if |i ? j| = 1. P_m × P_n will denote a graph whose vertices are elements of {1, 2, …, m} × {1, 2, …, n} and in which (i, j), (r, s) are adjacent whenever either i = r and |j ? s| = 1 or j = s and |i ? r| = 1.Theorem.If max(m, n) ? 2, thenφ(P_m × P_n) = min(m, n).     

A multi-dimensional Markov chain and the Meixner ensemble

We show that the transition probability of the Markov chain (G(i,1),...,G(i,n)) _i≥1, where the G(i,j)’s are certain directed last-passage times, is given by a determinant of a special form. An analogous formula has recently been obtained by Warren in a Brownian motion model. Furthermore we demonstrate that this formula leads to the Meixner ensemble when we compute the distribution function for G(m,n). We also obtain the Fredholm determinant representation of this distribution, where the kernel has a double contour integral representation.     

Induced matching extendable graphs

We say that a simple graph G is induced matching extendable, shortly IM-extendable, if every induced matching of G is included in a perfect matching of G. The main results of this paper are as follows: (1) For every connected IM-extendable graph G with |V(G)| ≥ 4, the girth g(G) ≤ 4. (2) If G is a connected IM-extendable graph, then |E(G)| ≥ ${3\over 2}|V(G)| - 2$; the equality holds if and only if G ≅ T × K₂, where T is a tree. (3) The only 3-regular connected IM-extendable graphs are C_n × K₂, for n ≥ 3, and C_2n(1, n), for n ≥ 2, where C_2n(1, n) is the graph with 2n vertices x₀, x₁, …, x_2n−1, such that x_ix_j is an edge of C_2n(1, n) if either |i − j| ≡ 1 (mod 2n) or |i − j| ≡ n (mod 2n). © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 203–213, 1998     

Hamiltonian cycles in 2-connected claw-free-graphs

M. Matthews and D. Sumner have proved that of G is a 2-connected claw-free graph of order n such that δ ≧ (n ? 2)/3, then G is hamiltonian. We prove that the bound for the minimum degree δ can be reduced to n/4 under the additional condition that G is not in F, where F is the set of all graphs defined as follows: any graph H in F can be decomposed into three vertex disjoint subgraphs H₁, H₂, H₃ such that , where u_i, v_i ? V(H_i), u_j v_j ? V(H_j) 1 ? i ≦ j ≦ 3. Examples are given to show that the bound n/4 is sharp. © 1995 John Wiley & Sons, Inc.     

Mixed ramsey numbers: Chromatic numbers versus graphs

For positive integers n₁, n₂, …, n_I and graphs G_I+1, G_I+2, …, G_k, 1 ≤ / < k, the mixed Ramsey number _χ(n₁, …, n₁, G_I+1, …, G_k) is define as the least positive integer p such that for each factorization K_p = F₁⊕ … ⊕ F_⊕ F_I+1⊕ … ⊕ F_k, it it follows that χ(F_i) ≥ n_i for some i, 1 ? i ? l, or G_i is a subgraph of F_i for some i, l < i ? k. Formulas are presented for maxed Ramsey numbers in which the graphs G_I+1, G_I+2, …, G_k are connected, and in which k = I+1 and G_I+1 is arbitray.     

On a class of transitive permutation groups of prime degreep=4n+1

SupposeG is a nonsolvable transitive permutation group of prime degreep, such that |N _G v(P)|=p(p−1) for some Sylowp-subgroupP ofG. Letq be a generator of the subgroup ofN _G (P), fixing one letter (it is easy to show that this subgroup is cyclic). Assume thatG contains an elementj such thatj ⁻¹ qj=q ^(p+1)/2. We shall prove that for almost all primesp of the formp=4n+1, a group that satisfies the above conditions must be the symmetric group on a set withp elements.     

Divisibility and structure constraints on automorphism groups of almost perfect 1-factorizations of

Let G be a permutation group acting on a set with N elements such that every permutation with more than m fixed points is the identity. It is easy to verify that |G| divides N(N − 1) ··· (N − m). We show that if gcd(|G|, m!) = 1, then |G| divides (N − i)(N − j) for some i and j satisfying 0 ≤ i < j ≤ m. We use this to show that any almost perfect 1-factorization of K_2n has an automorphism group whose cardinality divides (2n − i)(2n − j) for some i and j with 0 ≤ i < j ≤ 2 as long as n is odd. An almost perfect 1-factorization (or APOF) is a 1-factorization in which the union of any three distinct 1-factors is connected. This result contrasts with an example of an APOF on K₁₂ given by Cameron which has PSL(2, ℤ₁₁) as its automorphism group [with cardinality 12(11)(5)]. When n is even and the automorphism group is solvable, we show that either G acts vertex transitively and n is a power of two, or |G| divides 2n − 2^a for some integer a with 2^a dividing 2n, or else |G| divides (2n − i)(2n − j) for some i and j with 0 ≤ i < j ≤ 2. We also give a number of structure results concerning these automorphism groups. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 355–380, 1998     

On B(n,k) 2-groups

A finite group G is said to be a B(n, k) group if for any n-element subset {a ₁,…, a _n } of G, |{a _i a _j |1 ≤ i, j ≤ n}| ≤k. In this article, we give characterizations of the B(5, 19) 2-groups, and the B(6, k) 2-groups for 21 ≤ k ≤ 28.     

On the genus of a random graph

Let p = p(n) be a function of n with 0<p<1. We consider the random graph model ??(n, p); that is, the probability space of simple graphs with vertex-set {1, 2,…, n}, where two distinct vertices are adjacent with probability p. and for distinct pairs these events are mutually independent. Archdeacon and Grable have shown that if p²(1 ? p²) ?? 8(log n)⁴/n. then the (orientable) genus of a random graph in ??(n, p) is (1 + o(1))pn²/12. We prove that for every integer i ? 1, if n^?i/(i + 1) «p «n^{?(i ? 1)/i}. then the genus of a random graph in ??(n, p) is (1 + o(1))i/4(i + 2) pn². If p = cn^?(i?1)/o, where c is a constant, then the genus of a random graph in ??(n, p) is (1 + o(1))g(i, c, n)pn² for some function g(i, c, n) with 1/12 ? g(i, c, n) ? 1. but for i > 1 we were unable to compute this function.     

On bar frameworks,stress matrices and semidefinite programming

A bar framework G(p) in r-dimensional Euclidean space is a graph G on the vertices 1, 2, . . . , n, where each vertex i is located at point p ⁱ in \mathbbR^r{\mathbb{R}^r} . Given a framework G(p) in \mathbbR^r{\mathbb{R}^r} , a problem of great interest is that of determining whether or not there exists another framework G(q), not obtained from G(p) by a rigid motion, such that ||q ⁱ − q ^j ||² = ||p ⁱ − p ^j ||² for each edge (i, j) of G. This problem is known as either the global rigidity problem or the universal rigidity problem depending on whether such a framework G(q) is restricted to be in the same r-dimensional space or not. The stress matrix S of a bar framework G(p) plays a key role in these and other related problems. In this paper, semidefinite programming (SDP) theory is used to address, in a unified manner, several problems concerning universal rigidity. New results are presented as well as new proofs of previously known theorems. In particular, we use the notion of SDP non-degeneracy to obtain a sufficient condition for universal rigidity, and we show that this condition yields the previously known sufficient condition for generic universal rigidity. We present new results concerning positive semidefinite stress matrices and we use a semidefinite version of Farkas lemma to characterize bar frameworks that admit a nonzero positive semidefinite stress matrix S.     

On B(4, 13) 2-Groups

A group G is said to be a B(n, k) group if for any n-element subset {a ₁,…, a _n } of G, |{a _i a _j  | 1 ≤ i, j ≤ n}| ≤k. In this article, we give a complete characterization of B(4, 13) 2-groups, and then obtain a complete characterization of B(4, 13) groups.     

Hamiltonian Cycles in Almost Claw-Free Graphs

 Some known results on claw-free graphs are generalized to the larger class of almost claw-free graphs. In this paper, we prove the following two results and conjecture that every 5-connected almost claw-free graph is hamiltonian. (1). Every 2-connected almost claw-free graph G∉J on n≤ 4 δ vertices is hamiltonian, where J is the set of all graphs defined as follows: any graph G in J can be decomposed into three disjoint connected subgraphs G ₁, G ₂ and G ₃ such that E _G (G _i , G _j ) = {u _i , u _j , v _i v _j } for i≠j and i,j = 1, 2, 3 (where u _i ≠v _i ∈V(G _i ) for i = 1, 2, 3). Moreover the bound 4δ is best possible, thereby fully generalizing several previous results. (2). Every 3-connected almost claw-free graph on at most 5δ−5 vertices is hamiltonian, hereby fully generalizing the corresponding result on claw-free graphs. Received: September 21, 1998 Final version received: August 18, 1999     

1-Factorizations of cartesian products of regular graphs

Let G₁, G₂…, G_n be regular graphs and H be the Cartesian product of these graphs (H = G₁ × G₂ × … × G_n). The following will be proved: If the set {G₁, G₂…, G_n} has at leat one of the following properties: (*) for at leat one i ? {1, 2,…, n}, there exists a 1-factorization of G_i or (**) there exists at least two numbers i and j such that 1 ≤ i < j ≤ n and both the Graphs G_i and G_j contain at least one 1-factor, then there exists a 1-factorization of H. Further results: Let F be a cycle of length greater than three and let G be an arbitrary cubic graph. Then there exists a 1-factorization of the 5-regular graph H = F × G. The last result shows that neither (*) nor (**) is a necessary condition for the existence of a 1-factorization of a Cartesian product of regular graphs.     

Lower bounds for lower Ramsey numbers

For any graph G, let i(G) and μ;(G) denote the smallest number of vertices in a maximal independent set and maximal clique, respectively. For positive integers m and n, the lower Ramsey number s(m, n) is the largest integer p so that every graph of order p has i(G) ≤ m or μ;(G) ≤ n. In this paper we give several new lower bounds for s (m, n) as well as determine precisely the values s(1, n).     

OD-characterization of Almost Simple Groups Related to U3（5）

Let G be a finite group with order ｜G｜=p1^α1p2^α2……pk^αk, where p1 〈 p2 〈……〈 Pk are prime numbers. One of the well-known simple graphs associated with G is the prime graph （or Gruenberg- Kegel graph） denoted .by г（G） （or GK（G））. This graph is constructed as follows： The vertex set of it is π（G） = {p1,p2,…,pk} and two vertices pi, pj with i≠j are adjacent by an edge （and we write pi - pj） if and only if G contains an element of order pipj. The degree deg（pi） of a vertex pj ∈π（G） is the number of edges incident on pi. We define D（G） ：= （deg（p1）, deg（p2）,..., deg（pk））, which is called the degree pattern of G. A group G is called k-fold OD-characterizable if there exist exactly k non- isomorphic groups H such that ｜H｜ = ｜G｜ and D（H） = D（G）. Moreover, a 1-fold OD-characterizable group is simply called OD-characterizable. Let L ：= U3（5） be the projective special unitary group. In this paper, we classify groups with the same order and degree pattern as an almost simple group related to L. In fact, we obtain that L and L.2 are OD-characterizable; L.3 is 3-fold OD-characterizable; L.S3 is 6-fold OD-characterizable.     

Non-local Dirichlet forms generated by pseudodifferential operators on compact abelian groups

Let G_i, 1 ≤ i ≤ n, be compact abelian groups and let Γ_i , 1 ≤ i ≤ n, be countable dual groups. We consider G = G₁ ⊕ G₂ ⊕ … ⊕ G_n and Γ = Γ₁ ⊕ Γ₂ ⊕ … ⊕ Γ_n . For 1 ≤ j ≤ n, let a_j be a negative definite function on Γ_j and a (γ) = . For φ ∈ S (G), the set of all generalized trigonometrical polynomials on G, we define , where (γ) = a_j (γ_j) (γ), 1 ≤ j ≤ n. Then is a Dirichlet form with the domain on L² (G). (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on transitive permutation groups of prime degree

LetG be a nonsolvable transitive permutation group of prime degreep. LetP be a Sylow-p-subgroup ofG and letq be a generator of the subgroup ofN _G(P) fixing one point. Assume that |N _G(P)|=p(p?1) and that there exists an elementj inG such thatj ^?1qj=q^(p+1)/2. We shall prove that a group that satisfies the above condition must be the symmetric group onp points, andp is of the form 4n+1.     

Extremal results for rooted minor problems

In this article, we consider the following problem. Given four distinct vertices v₁,v₂,v₃,v₄. How many edges guarantee the existence of seven connected disjoint subgraphs X_i for i = 1,…, 7 such that X_j contains v_j for j = 1, 2, 3, 4 and for j = 1, 2, 3, 4, X_j has a neighbor to each X_k with k = 5, 6, 7. This is the so called “rooted K_{3, 4}‐minor problem.” There are only few known results on rooted minor problems, for example, [15,6]. In this article, we prove that a 4‐connected graph with n vertices and at least 5n ? 14 edges has a rooted K_3,4‐minor. In the proof we use a lemma on graphs with 9 vertices, proved by computer search. We also consider the similar problems concerning rooted K_3,3‐minor problem, and rooted K_3,2‐minor problem. More precisely, the first theorem says that if G is 3‐connected and e(G) ≥ 4|G| ? 9 then G has a rooted K_3,3‐minor, and the second theorem says that if G is 2‐connected and e(G) ≥ 13/5|G| ? 17/5 then G has a rooted K_3,2‐minor. In the second case, the extremal function for the number of edges is best possible. These results are then used in the proof of our forthcoming articles 7 , 8 . © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 191–207, 2007