The genus of the 2-amalgamations of graphs

A graph G is called the 2-amalgamation of subgraphs G₁ and G₂ if G = G₁ ∪ G₂ and G₁ ∩ G₂ = {x, y}, 2 distinct points. in this case we write G = G₁∪_{{x, y}} G₂. in this paper we show that the orientable genus, γ(G), satisfies the inequalities γ(G₁) + γ(G₂) ? 1 ≤ γ(G₁ ∪_{{x, y}} G₂) ≤ γ(G₁) + γ(G₂) + 1 and that this is the best possible result, i. e., the resulting three values for γ(G₁ ∪_{{x, y}} G₂) which are possible can actually be realized by appropriate choices for G₁ and G₂.     

Closed orbits and finite approximability with respect to conjugacy of free amalgamated products

We study the problem of finite approximability with respect to conjugacy of amalgamated free products by a normal subgroup and prove the following assertions. A) IfG is the amalgamated free productG=G _1*H G ₂ of polycyclic groupsG ₁ andG ₂ by a normal subgroupH, whereH is an almost free Abelian group of rank 2, thenG is finitely approximate with respect to conjugacy. B) (i) IfG ₁ =G ₂ =L is a polycyclic group andG=G _1*H G ₂ is the amalgamated product of two copies of the groupL by a normal subgroupH, thenG is finitely approximable with respect to conjugacy. (ii) IfG is an amalgamated free productG=G _1*H G ₂ of polycyclic groupsG ₁ andG ₂ by a normal subgroupH, whereH is central inG ₁ orG ₂, thenG is finitely approximable with respect to conjugacy.Translated fromMatematicheskie Zametki, Vol. 58, No. 4, pp. 525–535, October, 1995.This work was partially supported by the INTAS-94-3420 Grant Geometric Theory of Groups.     

Independence number,connectivity, and r-factors

We show that if r ? 1 is an odd integer and G is a graph with |V(G)| even such that k(G) ? (r + 1)²/2 and (r + 1)²α(G) ? 4rk(G), then G has an r-factor; if r ? 2 is even and G is a graph with k(G) ? r(r + 2)/2 and (r + 2)α(G) ? 4k(G), then G has an r-factor (where k(G) and α(G) denote the connectivity and the independence number of G, respectively).     

Retracts of box products with odd‐angulated factors

Let G be a connected graph with odd girth 2κ+1. Then G is a (2κ+1)‐angulated graph if every two vertices of G are connected by a path such that each edge of the path is in some (2κ+1)‐cycle. We prove that if G is (2κ+1)‐angulated, and H is connected with odd girth at least 2κ+3, then any retract of the box (or Cartesian) product G□ H is S□T where S is a retract of G and T is a connected subgraph of H. A graph G is strongly (2κ+1)‐angulated if any two vertices of G are connected by a sequence of (2κ+1)‐cycles with consecutive cycles sharing at least one edge. We prove that if G is strongly (2κ+1)‐angulated, and H is connected with odd girth at least 2κ+1, then any retract of G□ H is S□T where S is a retract of G and T is a connected subgraph of H or |V(S)|=1 and T is a retract of H. These two results improve theorems on weakly and strongly triangulated graphs by Nowakowski and Rival [Disc Math 70 ( 13 ), 169–184]. As a corollary, we get that the core of the box product of two strongly (2κ+1)‐angulated cores must be either one of the factors or the box product itself. Furthermore, if G is a strongly (2κ+1)‐angulated core, then either Gⁿ is a core for all positive integers n, or the core of Gⁿ is G for all positive integers n. In the latter case, G is homomorphically equivalent to a normal Cayley graph [Larose, Laviolette, Tardiff, European J Combin 19 ( 12 ), 867–881]. In particular, let G be a strongly (2κ+1)‐angulated core such that either G is not vertex‐transitive, or G is vertex‐transitive and any two maximum independent sets have non‐empty intersection. Then Gⁿ is a core for any positive integer n. On the other hand, let G_i be a (2κ_i+1)‐angulated core for 1 ≤ i≤ n where κ₁ < κ₂ < … < κ_n. If G_i has a vertex that is fixed under any automorphism for 1 ≤ i ≤ n‐1, or G_i is vertex‐transitive such that any two maximum independent sets have non‐empty intersection for 1 ≤ i ≤ n‐1, then □_i=1ⁿ G_i is a core. We then apply the results to construct cores that are box products with Mycielski construction factors or with odd graph factors. We also show that K(r,2r+1) □ C_2l+1 is a core for any integers l ≥ r ≥ 2. It is open whether K(r,2r+1) □ C_2l+1 is a core for r > l ≥ 2. © 2006 Wiley Periodicals, Inc. J Graph Theory     

The orientable genus is nonadditive

A graph G is a k-amalgamation of two graphs G₁ and G₂ if G = G₁ ∪ G₂ and G₁ ∩ G₂ is a set of k vertices. In this paper we construct 3-amalgamations G_n = H_n ∪ H_n such that γ(G_n) = 5n and γ(H_n) = 3n, where γ denotes the orientable genus of a graph. Thus γ(G₁ ∪ G₂) may differ from γ(G₁) + γ(G₂) by an arbitrarily large amount for amalgamations over 3 (or more) vertices. In contrast, an earlier paper shows that the nonorientable genus of a k-amalgamation differs from the sum of the nonorientable genera of its parts by at most a quadratic on k.     

On Two Theorems of Finite Solvable Groups

For a finite group G, let T（G） denote a set of primes such that a prime p belongs to T（G） if and only if p is a divisor of the index of some maximal subgroup of G. It is proved that if G satisfies any one of the following conditions： （1） G has a p-complement for each p∈T（G）; （2）│T（G）│= 2： （3） the normalizer of a Sylow p-subgroup of G has prime power index for each odd prime p∈T（G）; then G either is solvable or G/Sol（G）≌PSL（2, 7） where Sol（G） is the largest solvable normal subgroup of G.     

On a graph packing conjecture by Bollobás,Eldridge and Catlin

Two graphs G ₁ and G ₂ of order n pack if there exist injective mappings of their vertex sets into [n], such that the images of the edge sets are disjoint. In 1978, Bollobás and Eldridge, and independently Catlin, conjectured that if (Δ(G ₁) + 1)(Δ(G ₂) + 1) ≤ n + 1, then G ₁ and G ₂ pack. Towards this conjecture, we show that for Δ(G ₁),Δ(G ₂) ≥ 300, if (Δ(G ₁) + 1)(Δ(G ₂) + 1) ≤ 0.6n + 1, then G ₁ and G ₂ pack. This is also an improvement, for large maximum degrees, over the classical result by Sauer and Spencer that G ₁ and G ₂ pack if Δ(G ₁)Δ(G ₂) < 0.5n. This work was supported in part by NSF grant DMS-0400498. The work of the second author was also partly supported by NSF grant DMS-0650784 and grant 05-01-00816 of the Russian Foundation for Basic Research. The work of the third author was supported in part by NSF grant DMS-0652306.     

On Edge Singularity and Eigenvectors of Mixed Graphs

Let G be a mixed glaph which is obtained from an undirected graph by orienting some of its edges. The eigenvalues and eigenvectors of G are, respectively, defined to be those of the Laplacian matrix L（G） of G. As L（G） is positive semidefinite, the singularity of L（G） is determined by its least eigenvalue λ1 （G）. This paper introduces a new parameter edge singularity εs（G） that reflects the singularity of L（G）, which is the minimum number of edges of G whose deletion yields that all the components of the resulting graph are singular. We give some inequalities between εs（G） and λ1 （G） （and other parameters） of G. In the case of εs（G） = 1, we obtain a property on the structure of the eigenvectors of G corresponding to λ1 （G）, which is similar to the property of Fiedler vectors of a simple graph given by Fiedler.     

Overlap number of graphs

An overlap representation of a graph G assigns sets to vertices so that vertices are adjacent if and only if their assigned sets intersect with neither containing the other. The overlap number φ(G) (introduced by Rosgen) is the minimum size of the union of the sets in such a representation. We prove the following: (1) An optimal overlap representation of a tree can be produced in linear time, and its size is the number of vertices in the largest subtree in which the neighbor of any leaf has degree 2. (2) If δ(G)?2 and G≠K₃, then φ(G)?|E(G)| ? 1, with equality when G is connected and triangle‐free and has no star‐cutset. (3) If G is an n‐vertex plane graph with n?5, then φ(G)?2n ? 5, with equality when every face has length 4 and there is no star‐cutset. (4) If G is an n‐vertex graph with n?14, then φ(G)?n²/4 ? n/2 ? 1, with equality for even n when G arises from K_{n/2, n/2} by deleting a perfect matching. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Tilting modules for classical groups and howe duality in positive characteristics

We use the theory of tilting modules for algebraic groups to propose a characteristic free approach to “Howe duality” in the exterior algebra. To any series of classical groups (general linear, symplectic, orthogonal, or spinor) over an algebraically closed field k, we set in correspondence another series of classical groups (usually the same one). Denote byG ₁(m) the group of rankm from the first series and byG ₂(n) the group of rankn from the second series. For any pair (G ₁(m), G₂(n)) we construct theG ₁(m)×G₂(n)-module M(m, n). The construction of M(m, n) is independent of characteristic; for chark=0, the actions ofG ₁(m) andG ₂(n) on M(m, n) form a reductive dual pair in the sense of Howe. We prove that M(m, n) is a tiltingG ₁(m)-andG ₂(n)-module and that End _{G 1}(m) M(m, n) is generated byG ₂(n) and vice versa. The existence of such a module provides much information about the relations between the categoryK ₁(m, n) of rationalG ₁(m)-modules with highest weights bounded in a certain sense byn and the categoryK ₂(m, n) of rationalG ₂(n)-modules with highest weights bounded in the same sense bym. More specifically, we prove that there is a bijection of the set of dominant weights ofG ₁(m)-modules fromK ₁(m, n) to the set of dominant weights ofG ₂(m)-modules fromK ₂(m, n) such that Ext groups for inducedG ₁(m)-modules fromK ₁(m, n) are isomorphic to Ext groups for corresponding Weyl modules overG ₂(n). Moreover, the derived categoriesD ^bK₁(m, n) andD ^bK₂(m, n) appear to be equivalent. We also use our study of the modules M(m, n) to find generators and relations for the algebra of allG-invariants in , whereG=GL _m, Sp_2m, O_m and V is the naturalG-module. Research was supported in part by Grant M7N000/M7N300 from the International Science Foundation and Russian Government and by INTAS Grant 94-4720. Research was supported in part by Grant M8H000/M8H300 from the International Science Foundation and Russian Government and by INTAS Grant 94-4720.     

Linear arboricity for graphs with multiple edges

Akiyama, Exoo, and Harary conjectured that for any simple graph G with maximum degree Δ(G), the linear arboricity la(G) satisfies ?Δ(G)/2? ≦ la(G) ≦ ?(Δ(G) + 1)/2?. Here it is proved that if G is a loopless graph with maximum degree Δ(G) ≦ k and maximum edge multiplicity μ(G) ≦ k ? 2ⁿ⁺¹ + 1, where k ≧ 2^n?2, then la(G) ≦ k ? 2ⁿ. It is also conjectured that for any loopless graph G, ?Δ(G)/2? ≦ la(G) ≦ ?(Δ(G) + μ(G))/2?.     

On large sets of Pk‐decompositions

Let G = (V(G),E(G)) be a graph. A (ν, G, λ)‐GD is a partition of all the edges of λK_ν into subgraphs (G‐blocks), each of which is isomorphic to G. The (ν, G, λ)‐GD is named as graph design for G or G‐decomposition. The large set of (ν, G, λ)‐GD is denoted by (ν, G, λ)‐LGD. In this paper, we obtain a general result by using the finite fields, that is, if q ≥ k ≥ 2 is an odd prime power, then there exists a (q,P_k, k ? 1)‐LGD. © 2005 Wiley Periodicals, Inc. J Combin Designs.     

Computing the genus of the 2-amalgamations of graphs

The above authors [2] and S. Stahl [3] have shown that if a graphG is the 2-amalgamation of subgraphsG ₁ andG ₂ (namely ifG=G ₁∪G ₂ andG ₁∩G ₂={x, y}, two distinct points) then the orientable genus ofG,γ(G), is given byγ(G)=γ(G ₁)+γ(G ₂)+ε, whereε=0,1 or −1. In this paper we sharpen that result by giving a means by whichε may be computed exactly. This result is then used to give two irreducible graphs for each orientable surface.     

Bounds for the vertex linear arboricity

The vertex linear arboricity vla(G) of a nonempty graph G is the minimum number of subsets into which the vertex set V(G) can be partitioned so that each subset induces a subgraph whose connected components are paths. This paper provides an upper bound for vla(G) of a connected nonempty graph G, namely vla(G) ≦ 1 + ?δ(G)/2? where δ(G) denotes the maximum degree of G. Moreover, if δ(G) is even, then vla(G) = 1 + ?δ(G)/2? if and only if G is either a cycle or a complete graph.     

On the Interval Number of a Triangulated Graph

The interval number of a simple undirected graph G, denoted i(G), is the least nonnegative integer r for which we can assign to each vertex in G a collection of at most r intervals on the real line such that two distinct vertices v and w of G are adjacent if and only if some interval for v intersects some interval for w. For triangulated graphs G, we consider the problem of finding a sharp upper bound for the interval number of G in terms of its clique number ω(G). The following partial results are proved. (1) For each triangulated graph G, i(G) ? ?ω(G)/2? + 1, and this is best possible for 2 ? ω(G) ? 6. (2) For each integer m ? 2, there exists a triangulated graph G with ω(G) = m and i(G) > m^1/2.     

On packing 3‐vertex paths in a graph

Let G be a connected graph and let eb(G) and λ(G) denote the number of end‐blocks and the maximum number of disjoint 3‐vertex paths Λ in G. We prove the following theorems on claw‐free graphs: (t1) if G is claw‐free and eb(G) ≤ 2 (and in particular, G is 2‐connected) then λ(G) = ⌊| V(G)|/3⌋; (t2) if G is claw‐free and eb(G) ≥ 2 then λ(G) ≥ ⌊(| V(G) | − eb(G) + 2)/3 ⌋; and (t3) if G is claw‐free and Δ^*‐free then λ(G) = ⌊| V(G) |/3⌋ (here Δ^* is a graph obtained from a triangle Δ by attaching to each vertex a new dangling edge). We also give the following sufficient condition for a graph to have a Λ‐factor: Let n and p be integers, 1 ≤ p ≤ n − 2, G a 2‐connected graph, and |V(G)| = 3n. Suppose that G − S has a Λ‐factor for every S ⊆ V(G) such that |S| = 3p and both V(G) − S and S induce connected subgraphs in G. Then G has a Λ‐factor. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 175–197, 2001     

On a k-Tree Containing Specified Leaves in a Graph

A k-tree of a graph is a spanning tree with maximum degree at most k. We give sufficient conditions for a graph G to have a k-tree with specified leaves: Let k,s, and n be integers such that k≥2, 0≤s≤k, and n≥s+1. Suppose that (1) G is (s+1)-connected and the degree sum of any k independent vertices of G is at least |G|+(k−1)s−1, or (2) G is n-connected and the independence number of G is at most (n−s)(k−1)+1. Then for any s specified vertices of G, G has a k-tree containing them as leaves. We also discuss the sharpness of the results. This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists, 15740077, 2005 This research was partially supported by the Japan Society for the Promotion of Science for Young Scientists.     

Entropy,Periodicity, and Graphs with Zero Euler Characteristic

Let G be a graph which contains exactly one simple closed curve. We prove that a continuous map f : G → G has zero topological entropy if and only if there exist at most k ≤ |(Edg(G) End(G) 3)/2] different odd numbers n1,...,nk such that Per(f) is contained in ∪i=1^k ∪j=0^∞ ni2^j, where Edg(G) is the number of edges of G and End(G) is the number of end points of G.     

Recursive constructions for difference matrices and relative difference families

We present a new recursive construction for difference matrices whose application allows us to improve some results by D. Jungnickel. For instance, we prove that for any Abelian p-group G of type (n₁, n₂, …, n_t) there exists a (G, p^e, 1) difference matrix with e = Also, we prove that for any group G there exists a (G, p, 1) difference matrix where p is the smallest prime dividing |G|. Difference matrices are then used for constructing, recursively, relative difference families. We revisit some constructions by M. J. Colbourn, C. J. Colbourn, D. Jungnickel, K. T. Phelps, and R. M. Wilson. Combining them we get, in particular, the existence of a multiplier (G, k, λ)-DF for any Abelian group G of nonsquare-free order, whenever there exists a (p, k, λ)-DF for each prime p dividing |G|. Then we focus our attention on a recent construction by M. Jimbo. We improve this construction and prove, as a corollary, the existence of a (G, k, λ)-DF for any group G under the same conditions as above. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 165–182, 1998