r-Strong edge colorings of graphs

Let G be a graph and for any natural number r, denotes the minimum number of colors required for a proper edge coloring of G in which no two vertices with distance at most r are incident to edges colored with the same set of colors. In [Z. Zhang, L. Liu, J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15 (2002) 623-626] it has been proved that for any tree T with at least three vertices, . Here we generalize this result and show that . Moreover, we show that if for any two vertices u and v with maximum degree d(u,v)?3, then . Also for any tree T with Δ(T)?3 we prove that . Finally, it is shown that for any graph G with no isolated edges, .     

Hamilton decompositions of directed cubes and products

Call a directed graph symmetric if it is obtained from an undirected graph G by replacing each edge of G by two directed edges, one in each direction. We will show that if G has a Hamilton decomposition with certain additional structure, then has a directed Hamilton decomposition. In particular, it will follow that the bidirected cubes for m?2 are decomposable into 2m+1 directed Hamilton cycles and that a product of cycles is decomposable into 2m+1 directed Hamilton cycles if n_i?3 and m?2.     

Distance irredundance and connected domination numbers of a graph

Let k be a positive integer and G be a connected graph. This paper considers the relations among four graph theoretical parameters: the k-domination number γ_k(G), the connected k-domination number ; the k-independent domination number and the k-irredundance number ir_k(G). The authors prove that if an ir_k-set X is a k-independent set of G, then , and that for k?2, if ir_k(G)=1, if ir_k(G) is odd, and if ir_k(G) is even, which generalize some known results.     

Total domination and total domination subdivision number of a graph and its complement

A set S of vertices of a graph G=(V,E) with no isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination numberγ_t(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision numbersd_γt(G) is the minimum number of edges that must be subdivided in order to increase the total domination number. We consider graphs of order n?4, minimum degree δ and maximum degree Δ. We prove that if each component of G and has order at least 3 and , then and if each component of G and has order at least 2 and at least one component of G and has order at least 3, then . We also give a result on stronger than a conjecture by Harary and Haynes.     

On conjugate adjacency matrices of a graph

Let G be a simple graph of order n. Let and , where a and b are two nonzero integers and m is a positive integer such that m is not a perfect square. We say that A^c=[c_ij] is the conjugate adjacency matrix of the graph G if c_ij=c for any two adjacent vertices i and j, for any two nonadjacent vertices i and j, and c_ij=0 if i=j. Let P_G(λ)=|λI-A| and denote the characteristic polynomial and the conjugate characteristic polynomial of G, respectively. In this work we show that if then , where denotes the complement of G. In particular, we prove that if and only if P_G(λ)=P_H(λ) and . Further, let P^c(G) be the collection of conjugate characteristic polynomials of vertex-deleted subgraphs G_i=G?i(i=1,2,…,n). If P^c(G)=P^c(H) we prove that , provided that the order of G is greater than 2.     

More on decompositions of edge-colored complete graphs

Let G be a family of graphs whose edges are colored with elements from a set R of r colors. We assume no two vertices of G are joined by more than one edge of color i for any i∈R, for each G∈G. will denote the complete graph with r edges joining any pair of distinct vertices, one of each of the r colors. We describe necessary and asymptotically sufficient conditions on n for the existence of a family D of subgraphs of , each of which is an isomorphic copy of some graph in G, so that each edge of appears in exactly one of the subgraphs in D.     

Remarks on the minus (signed) total domination in graphs

A function f:V(G)→{+1,0,-1} defined on the vertices of a graph G is a minus total dominating function if the sum of its function values over any open neighborhood is at least 1. The minus total domination number of G is the minimum weight of a minus total dominating function on G. By simply changing “{+1,0,-1}” in the above definition to “{+1,-1}”, we can define the signed total dominating function and the signed total domination number of G. In this paper we present a sharp lower bound on the signed total domination number for a k-partite graph, which results in a short proof of a result due to Kang et al. on the minus total domination number for a k-partite graph. We also give sharp lower bounds on and for triangle-free graphs and characterize the extremal graphs achieving these bounds.     

The cubicity of hypercube graphs

For a graph G, its cubicity is the minimum dimension k such that G is representable as the intersection graph of (axis-parallel) cubes in k-dimensional space. (A k-dimensional cube is a Cartesian product R₁×R₂×?×R_k, where R_i is a closed interval of the form [a_i,a_i+1] on the real line.) Chandran et al. [L.S. Chandran, C. Mannino, G. Oriolo, On the cubicity of certain graphs, Information Processing Letters 94 (2005) 113-118] showed that for a d-dimensional hypercube H_d, . In this paper, we use the probabilistic method to show that . The parameter boxicity generalizes cubicity: the boxicity of a graph G is defined as the minimum dimension k such that G is representable as the intersection graph of axis-parallel boxes in k-dimensional space. Since for any graph G, our result implies that . The problem of determining a non-trivial lower bound for is left open.     

The subconstituent algebra of a bipartite distance-regular graph; thin modules with endpoint two

We consider a bipartite distance-regular graph Γ with diameter D?4, valency k?3, intersection numbers b_i,c_i, distance matrices A_i, and eigenvalues θ₀>θ₁>?>θ_D. Let X denote the vertex set of Γ and fix x∈X. Let T=T(x) denote the subalgebra of Mat_X(C) generated by , where A=A₁ and denotes the projection onto the ith subconstituent of Γ with respect to x. T is called the subconstituent algebra (or Terwilliger algebra) of Γ with respect to x. An irreducible T-module W is said to be thin whenever for 0?i?D. By the endpoint of W we mean . Assume W is thin with endpoint 2. Observe is a one-dimensional eigenspace for ; let η denote the corresponding eigenvalue. It is known where , and d=⌊D/2⌋. To describe the structure of W we distinguish four cases: (i) ; (ii) D is odd and ; (iii) D is even and ; (iv) . We investigated cases (i), (ii) in MacLean and Terwilliger [Taut distance-regular graphs and the subconstituent algebra, Discrete Math. 306 (2006) 1694-1721]. Here we investigate cases (iii), (iv) and obtain the following results. We show the dimension of W is D-1-e where e=1 in case (iii) and e=0 in case (iv). Let v denote a nonzero vector in . We show W has a basis , where E_i denotes the primitive idempotent of A associated with θ_i and where the set S is {1,2,…,d-1}∪{d+1,d+2,…,D-1} in case (iii) and {1,2,…,D-1} in case (iv). We show this basis is orthogonal (with respect to the Hermitian dot product) and we compute the square-norm of each basis vector. We show W has a basis , and we find the matrix representing A with respect to this basis. We show this basis is orthogonal and we compute the square-norm of each basis vector. We find the transition matrix relating our two bases for W.     

Spanning multi-paths in hypercubes

A set A of vertices of a hypercube is called balanced if . We prove that for every natural number n there exists a natural number π₁(n) such that for every hypercube Q with dim(Q)?π₁(n) there exists a family of pairwise vertex-disjoint paths P_i between A_i and B_i for i=1,2,…,n with if and only if {A_i,B_i∣i=1,2,…,n} is a balanced set.     

Long paths with endpoints in given vertex-subsets of graphs

Let G=(V,E) be a connected graph of order n, t a real number with t?1 and M⊆V(G) with . In this paper, we study the problem of some long paths to maintain their one or two different endpoints in M. We obtain the following two results: (1) for any vertex v∈V(G), there exists a vertex u∈M and a path P with the two endpoints v and u to satisfy , , d_G(u)+1-t}; (2) there exists either a cycle C to cover all vertices of M or a path P with two different endpoints u₀ and u_p in M to satisfy , where .     

The largest eigenvalue of unicyclic graphs

Let G be a simple graph. Let λ₁(G) and μ₁(G) denote the largest eigenvalue of the adjacency matrix and the Laplacian matrix of G, respectively. Let Δ denote the largest vertex degree. If G has just one cycle, then

     

Edges not contained in triangles and the number of contractible edges in a 4-connected graph

Let G be a 4-connected graph, and let E_c(G) denote the set of 4-contractible edges of G and let denote the set of those edges of G which are not contained in a triangle. Under this notation, we show that if , then we have .     

Quantifications des algèbres de Hopf d'arbres plans décorés et lien avec les groupes quantiques

We introduce a functor from the category of braided spaces into the category of braided Hopf algebras which associates to a braided space V a braided Hopf algebra of planar rooted trees . We show that the Nichols algebra of V is a subquotient of . We construct a Hopf pairing between and , generalising one of the results of [Bull. Sci. Math. 126 (2002) 193-239]. When the braiding of c is given by c(v_i⊗v_j)=q_i,jv_j⊗v_i, we obtain a quantification of the Hopf algebras introduced in [Bull. Sci. Math. 126 (2002) 193-239; 126 (2002) 249-288]. When q_i,j=q^a_i,j, with q an indeterminate and (a_i,j)_i,j the Cartan matrix of a semi-simple Lie algebra , then is a subquotient of . In this case, we construct the crossed product of with a torus and then the Drinfel'd quantum double of this Hopf algebra. We show that is a subquotient of .     

Lattices generated by join of strongly closed subgraphs in d-bounded distance-regular graphs

Let Γ be a d-bounded distance-regular graph with diameter d?3. Suppose that P(x) is a set of all strongly closed subgraphs containing x and that P(x,i) is a subset of P(x) consisting of all elements of P(x) with diameter i. Let L^′(x,i) be the set generated by all joins of the elements in P(x,i). By ordering L^′(x,i) by inclusion or reverse inclusion, L^′(x,i) is denoted by or . We prove that and are both finite atomic lattices, and give the conditions for them both being geometric lattices. We also give the eigenpolynomial of      

A contraction of square transversal designs

In this paper we show that if a square transversal design TD_λ[k;u], say D(=(P,B)), admits a class semiregular automorphism group G of order s, then we have a by matrix M with entries from G∪{0} satisfying , where , if i=j, and , otherwise. As an application of (*), we show that any symmetric TD₂[12;6] admits no nontrivial elation. We also obtain a result that gives us a restriction on the existence of elations of putative projective planes of composite order.     

The asymptotic number of spanning trees in circulant graphs

Let T(G) be the number of spanning trees in graph G. In this note, we explore the asymptotics of T(G) when G is a circulant graph with given jumps.The circulant graph is the 2k-regular graph with n vertices labeled 0,1,2,…,n−1, where node i has the 2k neighbors i±s₁,i±s₂,…,i±s_k where all the operations are . We give a closed formula for the asymptotic limit as a function of s₁,s₂,…,s_k. We then extend this by permitting some of the jumps to be linear functions of n, i.e., letting s_i, d_i and e_i be arbitrary integers, and examining

     

Total restrained domination in graphs with minimum degree two

In this paper, we continue the study of total restrained domination in graphs, a concept introduced by Telle and Proskurowksi (Algorithms for vertex partitioning problems on partial k-trees, SIAM J. Discrete Math. 10 (1997) 529-550) as a vertex partitioning problem. A set S of vertices in a graph G=(V,E) is a total restrained dominating set of G if every vertex is adjacent to a vertex in S and every vertex of V?S is adjacent to a vertex in V?S. The minimum cardinality of a total restrained dominating set of G is the total restrained domination number of G, denoted by γ_tr(G). Let G be a connected graph of order n with minimum degree at least 2 and with maximum degree Δ where Δ?n-2. We prove that if n?4, then and this bound is sharp. If we restrict G to a bipartite graph with Δ?3, then we improve this bound by showing that and that this bound is sharp.     

A note on edge-graceful spectra of the square of paths

For a simple path P_r on r vertices, the square of P_r is the graph on the same set of vertices of P_r, and where every pair of vertices of distance two or less in P_r is connected by an edge. Given a (p,q)-graph G with p vertices and q edges, and a nonnegative integer k, G is said to be k-edge-graceful if the edges can be labeled bijectively by k,k+1,…,k+q−1, so that the induced vertex sums are pairwise distinct, where the vertex sum at a vertex is the sum of the labels of all edges incident to such a vertex, modulo the number of vertices p. We call the set of all such k the edge-graceful spectrum of G, and denote it by egI(G). In this article, the edge-graceful spectrum for the square of paths is completely determined for odd r.     

Upper minus domination in a claw-free cubic graph

We show in this paper that the upper minus domination number Γ^-(G) of a claw-free cubic graph G is at most .