Cube intersection concepts in median graphs

In this paper, we study different classes of intersection graphs of maximal hypercubes of median graphs. For a median graph G and k≥0, the intersection graph Q_k(G) is defined as the graph whose vertices are maximal hypercubes (by inclusion) in G, and two vertices H_x and H_y in Q_k(G) are adjacent whenever the intersection H_x∩H_y contains a subgraph isomorphic to Q_k. Characterizations of clique-graphs in terms of these intersection concepts when k>0, are presented. Furthermore, we introduce the so-called maximal 2-intersection graph of maximal hypercubes of a median graph G, denoted , whose vertices are maximal hypercubes of G, and two vertices are adjacent if the intersection of the corresponding hypercubes is not a proper subcube of some intersection of two maximal hypercubes. We show that a graph H is diamond-free if and only if there exists a median graph G such that H is isomorphic to . We also study convergence of median graphs to the one-vertex graph with respect to all these operations.     

A characterization of the interval distance monotone graphs

A simple connected graph G is said to be interval distance monotone if the interval I(u,v) between any pair of vertices u and v in G induces a distance monotone graph. A?¨der and Aouchiche [Distance monotonicity and a new characterization of hypercubes, Discrete Math. 245 (2002) 55-62] proposed the following conjecture: a graph G is interval distance monotone if and only if each of its intervals is either isomorphic to a path or to a cycle or to a hypercube. In this paper we verify the conjecture.     

Distorsion des distances dans les groupes de Lie nilpotents

Let G be a connected nilpotent Lie group and H a connected subgroup of G. We give an explicit formula for the distance to the origin with the exponential coordinates of the second kind of g∈G. Using this fact, we prove that the distance to the origin of any element in H is bounded by a polynomial function of the distance to the origin in the group G. The degree of the polynomial is the nilpotency rank of the group G.     

A Sharp Bound for the Marginal Appendage Number

For a connected graph G, the distance d(u, v) between two vertices u and v is the length of a shortest u − v path in G and the distance d(v) of a vertex v is the sum of the distances between v and all vertices of G. The margin, μ (G), is the subgraph induced by vertices of G having the maximum distance. It is known that every graph is isomorphic to the margin of some graph H. For a graph G, the marginal appendage number is defined as min{p(H) − p(G) ∣ μ(H) = G}. In this paper it is shown that Δ (G) + 2 is a sharp bound for the marginal appendage number.     

On the remoteness function in median graphs

A profile on a graph G is any nonempty multiset whose elements are vertices from G. The corresponding remoteness function associates to each vertex x∈V(G) the sum of distances from x to the vertices in the profile. Starting from some nice and useful properties of the remoteness function in hypercubes, the remoteness function is studied in arbitrary median graphs with respect to their isometric embeddings in hypercubes. In particular, a relation between the vertices in a median graph G whose remoteness function is maximum (antimedian set of G) with the antimedian set of the host hypercube is found. While for odd profiles the antimedian set is an independent set that lies in the strict boundary of a median graph, there exist median graphs in which special even profiles yield a constant remoteness function. We characterize such median graphs in two ways: as the graphs whose periphery transversal number is 2, and as the graphs with the geodetic number equal to 2. Finally, we present an algorithm that, given a graph G on n vertices and m edges, decides in O(mlogn) time whether G is a median graph with geodetic number 2.     

Fall colouring of bipartite graphs and cartesian products of graphs

The question of whether a graph can be partitioned into k independent dominating sets, which is the same as having a fallk-colouring, is considered. For k=3, it is shown that a graph G can be partitioned into three independent dominating sets if and only if the cartesian product G□K₂ can be partitioned into three independent dominating sets. The graph K₂ can be replaced by any graph H such that there is a mapping f:Q_n→H, where f is a type-II graph homomorphism.The cartesian product of two trees is considered, as well as the complexity of partitioning a bipartite graph into three independent dominating sets, which is shown to be NP-complete. For other values of k, iterated cartesian products are considered, leading to a result that shows for what values of k the hypercubes can be partitioned into k independent dominating sets.     

On spanning tree congestion of graphs

Let G be a connected graph and T be a spanning tree of G. For e∈E(T), the congestion of e is the number of edges in G connecting two components of T−e. The edge congestion ofGinT is the maximum congestion over all edges in T. The spanning tree congestion ofG is the minimum congestion of G in its spanning trees. In this paper, we show the spanning tree congestion for the complete k-partite graphs and the two-dimensional tori. We also address lower bounds of spanning tree congestion for the multi-dimensional grids and the hypercubes.     

The Hosoya polynomial decomposition for catacondensed benzenoid graphs

For a graph G with the vertex set V(G), we denote by d(u,v) the distance between vertices u and v in G, by d(u) the degree of vertex u. The Hosoya polynomial of G is H(G)=∑_{u,v}⊆V(G)x^d(u,v). The partial Hosoya polynomials of G are for positive integer numbers m and n. It is shown that H(G₁)−H(G₂)=x²(x+1)²(H₃₃(G₁)−H₃₃(G₂)),H₂₂(G₁)−H₂₂(G₂)=(x²+x−1)²(H₃₃(G₁)−H₃₃(G₂)) and H₂₃(G₁)−H₂₃(G₂)=2(x²+x−1)(H₃₃(G₁)−H₃₃(G₂)) for arbitrary catacondensed benzenoid graphs G₁ and G₂ with equal number of hexagons. As an application, we give an affine relationship between H(G) with two other distance-based polynomials constructed by Gutman [I. Gutman, Some relations between distance-based polynomials of trees, Bulletin de l’Académie Serbe des Sciences et des Arts (Cl. Math. Natur.) 131 (2005) 1-7].     

A note on the partition dimension of Cartesian product graphs

Let G = (V, E) be a connected graph. The distance between two vertices u, v ∈ V, denoted by d(u, v), is the length of a shortest u − v path in G. The distance between a vertex v ∈ V and a subset P ⊂ V is defined as , and it is denoted by d(v, P). An ordered partition {P₁, P₂, … , P_t} of vertices of a graph G, is a resolving partition of G, if all the distance vectors (d(v, P₁), d(v, P₂), … , d(v, P_t)) are different. The partition dimension of G, denoted by pd(G), is the minimum number of sets in any resolving partition of G. In this article we study the partition dimension of Cartesian product graphs. More precisely, we show that for all pairs of connected graphs G, H, pd(G × H) ? pd(G) + pd(H) and pd(G × H) ? pd(G) + dim(H), where dim(H) denotes the metric dimension of H. Consequently, we show that pd(G × H) ? dim(G) + dim(H) + 1.     

On <Emphasis Type="Italic">k</Emphasis>-pairable graphs from trees

The concept of the k-pairable graphs was introduced by Zhibo Chen (On k-pairable graphs, Discrete Mathematics 287 (2004), 11–15) as an extension of hypercubes and graphs with an antipodal isomorphism. In the same paper, Chen also introduced a new graph parameter p(G), called the pair length of a graph G, as the maximum k such that G is k-pairable and p(G) = 0 if G is not k-pairable for any positive integer k. In this paper, we answer the two open questions raised by Chen in the case that the graphs involved are restricted to be trees. That is, we characterize the trees G with p(G) = 1 and prove that p(G □ H) = p(G) + p(H) when both G and H are trees.     

Independence and coloring properties of direct products of some vertex-transitive graphs

Let α(G) and χ(G) denote the independence number and chromatic number of a graph G, respectively. Let G×H be the direct product graph of graphs G and H. We show that if G and H are circular graphs, Kneser graphs, or powers of cycles, then α(G×H)=max{α(G)|V(H)|,α(H)|V(G)|} and χ(G×H)=min{χ(G),χ(H)}.     

On permuteral subgroups in finite groups

The permutizer of a subgroup H in a group G is defined as the subgroup generated by all cyclic subgroups of G that permute with H. Call H permuteral in G if the permutizer of H in G coincides with G; H is called strongly permuteral in G if the permutizer of H in U coincides with U for every subgroup U of G containing H. We study the finite groups with given systems of permuteral and strongly permuteral subgroups and find some new characterizations of w-supersoluble and supersoluble groups.     

Subgraphs and the Laplacian spectrum of a graph

Let G be a graph and H a subgraph of G. In this paper, a set of pairwise independent subgraphs that are all isomorphic copies of H is called an H-matching. Denoting by ν(H,G) the cardinality of a maximum H-matching in G, we investigate some relations between ν(H,G) and the Laplacian spectrum of G.     

A common cover of graphs and 2-cell embeddings

Let G and H be finite graphs with equal uniform degree refinements. Their finite common covering graph G∘H is constructed. It is shown that G, H, and G∘H can be 2-cell embedded in orientable surfaces M, N and S^∼, respectively, in such a way that the graph covering projections G∘H → G and G∘H → H extend to branched coverings M ← S^∼ → N of the surfaces. Additional properties of G∘H are used to obtain some nontrivial consequences about coverings of some planar graphs.     

Relative colorings of graphs

In this paper, the notion of relative chromatic number χ(G, H) for a pair of graphs G, H, with H a full subgraph of G, is formulated; namely, χ(G, H) is the minimum number of new colors needed to extend any coloring of H to a coloring of G. It is shown that the four color conjecture (4CC) is equivalent to the conjecture (R4CC) that χ(G, H) ≤ 4 for any (possibly empty) full subgraph H of a planar graph G and also to the conjecture (CR3CC) that χ(G, H) ≤ 3 if H is a connected and nonempty full subgraph of planar G. Finally, relative coloring theorems on surfaces other than the plane or sphere are proved.     

Eigenvalues of a graph and its imbeddings

A graph H is imbedded in a graph G if a subset of the vertices of G determines a subgraph isomorphic to H. If λ(G) is the least eigenvalue of G and k_R(H) = lim sup_d→∞ {λ(G)| H imbedded in G; G regular and connected; diam(G) > d; deg(G) > d}, then λ(H) ? 2 ≤ k_R(H) ≤ λ(H) with these bounds being the best possible. Given a graph H, there exist arbitrarily large families of isospectral graphs such that H can be imbedded in each member of the family.     

Finite groups with some maximal subgroups of Sylow subgroups ℳ-supplemented

A subgroup H of a group G is said to be ?-supplemented in G if there exists a subgroup B of G such that G = HB and TB < G for every maximal subgroup T of H. In this paper, we obtain the following statement: Let ? be a saturated formation containing all supersolvable groups and H be a normal subgroup of G such that G/H ε ?. Suppose that every maximal subgroup of a noncyclic Sylow subgroup of F*(H), having no supersolvable supplement in G, is ?-supplemented in G. Then G ε ?.     

Finite groups with given nearly s-embedded subgroups

Let G be a finite group and H a subgroup of G. We say that H is s-permutable in G if HP = PH for all Sylow subgroups P of G; H is s-semipermutable in G if HP = PH for all Sylow subgroups P of G with (|P|, |H|) = 1. Let H _s G be the subgroup of H generated by all those subgroups of G which are s-permutable in G and H ^sG the intersection of all such s-permutable subgroups of G contain H. We say that H is nearly s-embedded in G if G has an s-permutable subgroup T such that H ^sG = HT and \({H \cap T \leqq H_{ssG}}\) , where H _ssG is an s-semipermutable subgroup of G contained in H. In this paper, we study the structure of a finite group G under the assumption that some subgroups of prime power order are nearly s-embedded in G. A series of known results are improved and extended.     

Spherical homogeneous spaces of minimal rank

Let G be a connected reductive algebraic group over an algebraically closed field K of characteristic zero. Let G/B denote the complete flag variety of G. A G-homogeneous space G/H is said to be spherical if H has finitely many orbits in G/B. A class of spherical homogeneous spaces containing the tori, the complete homogeneous spaces and the group G (viewed as a G×G-homogeneous space) has particularly nice properties. Namely, the pair (G,H) is called a spherical pair of minimal rank if there exists x in G/B such that the orbit H.x of x by H is open in G/B and the stabilizer Hx of x in H contains a maximal torus of H. In this article, we study and classify the spherical pairs of minimal rank.     

On Nearly SS-Embedded Subgroups of Finite Groups

Let H be a subgroup of a finite group G. H is nearly SS-embedded in G if there exists an S-quasinormal subgroup K of G, such that HK is S-quasinormal in G and H ∩ K ≤ HseG, where HseG is the subgroup of H, generated by all those subgroups of H which are S-quasinormally embedded in G. In this paper, the authors investigate the influence of nearly SS-embedded subgroups on the structure of finite groups.