Graphs satisfying inequality θ(G)θ(G)

In this paper, we study the edge clique cover number of squares of graphs. More specifically, we study the inequality θ(G²)θ(G) where θ(G) is the edge clique cover number of a graph G. We show that any graph G with at most θ(G) vertices satisfies the inequality. Among the graphs with more than θ(G) vertices, we find some graphs violating the inequality and show that dually chordal graphs and power-chordal graphs satisfy the inequality. Especially, we give an exact formula computing θ(T²) for a tree T.     

Enumeration of perfect matchings of a type of Cartesian products of graphs

Let G be a graph and let Pm(G) denote the number of perfect matchings of G.We denote the path with m vertices by P_m and the Cartesian product of graphs G and H by G×H. In this paper, as the continuance of our paper [W. Yan, F. Zhang, Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians, Adv. Appl. Math. 32 (2004) 175-188], we enumerate perfect matchings in a type of Cartesian products of graphs by the Pfaffian method, which was discovered by Kasteleyn. Here are some of our results:1. Let T be a tree and let C_n denote the cycle with n vertices. Then Pm(C₄×T)=∏(2+α²), where the product ranges over all eigenvalues α of T. Moreover, we prove that Pm(C₄×T) is always a square or double a square.2. Let T be a tree. Then Pm(P₄×T)=∏(1+3α²+α⁴), where the product ranges over all non-negative eigenvalues α of T.3. Let T be a tree with a perfect matching. Then Pm(P₃×T)=∏(2+α²), where the product ranges over all positive eigenvalues α of T. Moreover, we prove that Pm(C₄×T)=[Pm(P₃×T)]².     

The rainbow number of matchings in regular bipartite graphs

Given a graph G and a subgraph H of G, let rb(G,H) be the minimum number r for which any edge-coloring of G with r colors has a rainbow subgraph H. The number rb(G,H) is called the rainbow number of H with respect to G. Denote as mK₂ a matching of size m and as B_n,k the set of all the k-regular bipartite graphs with bipartition (X,Y) such that X=Y=n and k≤n. Let k,m,n be given positive integers, where k≥3, m≥2 and n>3(m−1). We show that for every GB_n,k, rb(G,mK₂)=k(m−2)+2. We also determine the rainbow numbers of matchings in paths and cycles.     

On graphs determining links with maximal number of components via medial construction

Let G be a connected plane graph, D(G) be the corresponding link diagram via medial construction, and μ(D(G)) be the number of components of the link diagram D(G). In this paper, we first provide an elementary proof that μ(D(G))≤n(G)+1, where n(G) is the nullity of G. Then we lay emphasis on the extremal graphs, i.e. the graphs with μ(D(G))=n(G)+1. An algorithm is given firstly to judge whether a graph is extremal or not, then we prove that all extremal graphs can be obtained from K₁ by applying two graph operations repeatedly. We also present a dual characterization of extremal graphs and finally we provide a simple criterion on structures of bridgeless extremal graphs.     

Matchings and spanning trees in Boolean weighted graphs

It is shown that in a 0-sum Boolean weighted graph G the sum of the weights taken over all the spanning trees equals the sum of the weights taken over all the perfect matchings in the graph G − v, where v is any vertex of G. Several related theorems are proved which include parity results on perfect matchings and spanning trees in Eulerian graphs. The ideas on perfect matchings in 0-sum Boolean weighted graphs are generalized to matchings in any Boolean weighted graph.     

Average distance,minimum degree,and spanning trees

The average distance μ(G) of a connected graph G of order n is the average of the distances between all pairs of vertices of G, i.e., μ(G) = ()⁻¹ Σ_{x,y}⊂V(G) d_G(x, y), where V(G) denotes the vertex set of G and d_G(x, y) is the distance between x and y. We prove that every connected graph of order n and minimum degree δ has a spanning tree T with average distance at most . We give improved bounds for K₃‐free graphs, C₄‐free graphs, and for graphs of given girth. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 1–13, 2000     

Continuity in G

For a discrete group G, we consider βG, the Stone– ech compactification of G, as a right topological semigroup, and G^*=βGG as a subsemigroup of βG. We study the mappings λ_p^* :G^*→G^*and μ^* :G^*→G^*, the restrictions to G^* of the mappings λ_p :βG→βG and μ :βG→βG, defined by the rules λ_p(q)=pq, μ(q)=qq. Under some assumptions, we prove that the continuity of λ_p^* or μ^* at some point of G^* implies the existence of a P-point in ω^*.     

Vertex Set Partitions Preserving Conservativeness

Let G be an undirected graph and ={X₁, …, X_n} be a partition of V(G). Denote by G/ the graph which has vertex set {X₁, …, X_n}, edge set E, and is obtained from G by identifying vertices in each class X_i of the partition . Given a conservative graph (G, w), we study vertex set partitions preserving conservativeness, i.e., those for which (G/ , w) is also a conservative graph. We characterize the conservative graphs (G/ , w), where is a terminal partition of V(G) (a partition preserving conservativeness which is not a refinement of any other partition of this kind). We prove that many conservative graphs admit terminal partitions with some additional properties. The results obtained are then used in new unified short proofs for a co-NP characterization of Seymour graphs by A. A. Ageev, A. V. Kostochka, and Z. Szigeti (1997, J. Graph Theory34, 357–364), a theorem of E. Korach and M. Penn (1992, Math. Programming55, 183–191), a theorem of E. Korach (1994, J. Combin. Theory Ser. B62, 1–10), and a theorem of A. V. Kostochka (1994, in “Discrete Analysis and Operations Research. Mathematics and its Applications (A. D. Korshunov, Ed.), Vol. 355, pp. 109–123, Kluwer Academic, Dordrecht).     

Commuting decompositions of complete graphs

We say that two graphs G and H with the same vertex set commute if their adjacency matrices commute. In this article, we show that for any natural number r, the complete multigraph K is decomposable into commuting perfect matchings if and only if n is a 2‐power. Also, it is shown that the complete graph K_n is decomposable into commuting Hamilton cycles if and only if n is a prime number. © 2006 Wiley Periodicals, Inc. J Combin Designs     

Affine Hecke algebras and the Schubert calculus

Using a combinatorial approach that avoids geometry, this paper studies the structure of K_T(G/B), the T-equivariant K-theory of the generalized flag variety G/B. This ring has a natural basis (the double Grothendieck polynomials), where is the structure sheaf of the Schubert variety X_w. For rank two cases we compute the corresponding structure constants of the ring K_T(G/B) and, based on this data, make a positivity conjecture for general G which generalizes the theorems of M. Brion (for K(G/B)) and W. Graham (for H_T^*(G/B)). Let [X^λ]K_T(G/B) be the class of the homogeneous line bundle on G/B corresponding to the character of T indexed by λ. For general G we prove “Pieri–Chevalley formulas” for the products , , , and , where λ is dominant. By using the Chern character and comparing lowest degree terms the products which are computed in this paper also give results for the Grothendieck polynomials, double Schubert polynomials, and ordinary Schubert polynomials in, respectively K(G/B), H_T^*(G/B) and H^*(G/B).     

Equivariant K-Homology and Restriction to Finite Cyclic Subgroups

For a discrete group G, we prove that a G-map between proper G–CW-complexes induces an isomorphism in G-equivariant K-homology if it induces an isomorphism in C-equivariant K-homology for every finite cyclic subgroup C of G. As an application, we show that the source of the Baum–Connes assembly map, namely K _* ^G (E(G, in)), is isomorphic to K _* ^G (E(G, )), where E(G, ) denotes the classifying space for the family of finite cyclic subgroups of G. Letting be the family of virtually cyclic subgroups of G, we also establish that and related results.     

Adjacent strong edge colorings and total colorings of regular graphs

It is conjectured that χas(G) = χt(G) for every k-regular graph G with no C5 component (k 2). This conjecture is shown to be true for many classes of graphs, including: graphs of type 1; 2-regular, 3-regular and (|V (G)| - 2)-regular graphs; bipartite graphs; balanced complete multipartite graphs; k-cubes; and joins of two matchings or cycles.     

On <Emphasis Type="Italic">k</Emphasis>-th Power of Upper Bound Graphs

We consider k-th power of upper bound graphs. According to the characterization of upper bound graphs, we obtain a characterization of k-th power of upper bound graphs. That is, for a connected upper bound graph G, G^k is an upper bound graph if and only if for any pair of A^k -simplicial vertices s₁, s₂ such that , there exists a G^k -simplicial vertex s satisfying the conditions: and . Furthermore we also get some properties on squares of upper bound graphs.AMS Subject Classification: 05C62.     

Bounded edge-connectivity and edge-persistence of Cartesian product of graphs

The bounded edge-connectivity λ_k(G) of a connected graph G with respect to is the minimum number of edges in G whose deletion from G results in a subgraph with diameter larger than k and the edge-persistence D⁺(G) is defined as λ_d(G)(G), where d(G) is the diameter of G. This paper considers the Cartesian product G₁×G₂, shows λ_k1+k2(G₁×G₂)≥λ_k1(G₁)+λ_k2(G₂) for k₁≥2 and k₂≥2, and determines the exact values of D⁺(G) for G=C_n×P_m, C_n×C_m, Q_n×P_m and Q_n×C_m.     

De Bruijn digraphs and affine transformations

Let be the additive group of 1×n row vectors over . For an n×n matrix T over  and , the affine transformation F_T,ω of sends x to xT+ω. Let α be the cyclic group generated by a vector . The affine transformation coset pseudo-digraph has the set of cosets of α in as vertices and there are c arcs from x+α to y+α if and only if the number of zx+α such that F_T,ω(z)y+α is c. We prove that the following statements are equivalent: (a)  is isomorphic to the d-nary (n−1)-dimensional De Bruijn digraph; (b) α is a cyclic vector for T; (c)  is primitive. This strengthens a result conjectured by C.M. Fiduccia and E.M. Jacobson [Universal multistage networks via linear permutations, in: Proceedings of the 1991 ACM/IEEE Conference on Supercomputing, ACM Press, New York, 1991, pp. 380–389]. Under the further assumption that T is invertible we show that each component of is a conjunction of a cycle and a De Bruijn digraph, namely a generalized wrapped butterfly. Finally, we discuss the affine TCP digraph representations for a class of digraphs introduced by D. Coudert, A. Ferreira and S. Perennes [Isomorphisms of the De Bruijn digraph and free-space optical networks, Networks 40 (2002) 155–164].     

Coloring of distance graphs with intervals as distance sets

Let D be a set of positive integers. The distance graph G(Z,D) with distance set D is the graph with vertex set Z in which two vertices x,y are adjacent if and only if |x−y|D. The fractional chromatic number, the chromatic number, and the circular chromatic number of G(Z,D) for various D have been extensively studied recently. In this paper, we investigate the fractional chromatic number, the chromatic number, and the circular chromatic number of the distance graphs with the distance sets of the form D_m,[k,k^′]={1,2,…,m}−{k,k+1,…,k^′}, where m, k, and k^′ are natural numbers with m≥k^′≥k. In particular, we completely determine the chromatic number of G(Z,D_m,[2,k^′]) for arbitrary m, and k^′.     

On hamiltonicity of <Emphasis Type="Italic">P</Emphasis><Subscript>3</Subscript>-dominated graphs

We introduce a new class of graphs which we call P ₃-dominated graphs. This class properly contains all quasi-claw-free graphs, and hence all claw-free graphs. Let G be a 2-connected P ₃-dominated graph. We prove that G is hamiltonian if α(G ²) ≤ κ(G), with two exceptions: K _2,3 and K _1,1,3. We also prove that G is hamiltonian, if G is 3-connected and |V(G)| ≤ 5δ(G) − 5. These results extend known results on (quasi-)claw-free graphs. This paper was completed when both authors visited the Center for Combinatorics, Nankai University, Tianjin. They gratefully acknowledge the hospitality and support of the Center for Combinatorics and Nankai University. The work of E.Vumar is sponsored by SRF for ROCS, REM.     

Signed Total Domination Nnumber of a Graph

The signed total domination number of a graph is a certain variant of the domination number. If is a vertex of a graph G, then N() is its oper neighbourhood, i.e. the set of all vertices adjacent to in G. A mapping f: V(G)-1, 1, where V(G) is the vertex set of G, is called a signed total dominating function (STDF) on G, if for each V(G). The minimum of values , taken over all STDF's of G, is called the signed total domination number of G and denoted by _st(G). A theorem stating lower bounds for _st(G) is stated for the case of regular graphs. The values of this number are found for complete graphs, circuits, complete bipartite graphs and graphs on n-side prisms. At the end it is proved that _st(G) is not bounded from below in general.     

The Faber Operator and its Boundedness

Let G be a domain bounded by a Jordan curve Γ, and let A(G) be the Banach space of functions continuous on G and holomorphic in G. The Faber operator T is a linear mapping from A( ) to A(G) mapping wⁿ onto the nth Faber polynomial F_n(z) (n=0, 1, 2, …). We show that T<∞ if Γ is piecewise Dini-smooth, and give an example of a quasicircle Γ for which T=∞.     

Forcing matching numbers of fullerene graphs

The forcing number or the degree of freedom of a perfect matching M of a graph G is the cardinality of the smallest subset of M that is contained in no other perfect matchings of G. In this paper we show that the forcing numbers of perfect matchings in a fullerene graph are not less than 3 by applying the 2-extendability and cyclic edge-connectivity 5 of fullerene graphs obtained recently, and Kotzig’s classical result about unique perfect matching as well. This lower bound can be achieved by infinitely many fullerene graphs.