The bandwidth sum of join and composition of graphs

Given a graph G, a proper labeling f of G is a one-to-one function . The bandwidth sum of a graph G, denoted by B_s(G), is defined by B_s(G)=min∑_uvE(G)|f(u)-f(v)|, where the minimum is taken for all proper labelings f of G. In this paper, we give some results for the bandwidth sum problem for the join of k graphs G₁,G₂,…,G_k, where each G_i is a path, cycle, complete graph, or union of isolated vertices. We also discuss the bandwidth sum for the composition of two graphs G and H, where G and H are path, cycle, or union of isolated vertices.     

All triangle-graph ramsey numbers for connected graphs of order six

The Ramsey numbers r(K_3′ G) are determined for all connected graphs G of order six.     

A strong law of large numbers for random biased connected graphs

We consider a class of random connected graphs with random vertices and random edges in which the randomness of the vertices is determined by a continuous probability distribution and the randomness of the edges is determined by a connection function. We derive a strong law of large numbers on the total lengths of all random edges for a random biased connected graph that is a generalization of a directed k-nearest-neighbor graph.     

Decompositions for edge-coloring join graphs and cobipartite graphs

An edge-coloring is an association of colors to the edges of a graph, in such a way that no pair of adjacent edges receive the same color. A graph G is Class 1 if it is edge-colorable with a number of colors equal to its maximum degree Δ(G). To determine whether a graph G is Class 1 is NP-complete [I. Holyer, The NP-completeness of edge-coloring, SIAM J. Comput. 10 (1981) 718-720]. First, we propose edge-decompositions of a graph G with the goal of edge-coloring G with Δ(G) colors. Second, we apply these decompositions for identifying new subsets of Class 1 join graphs and cobipartite graphs. Third, the proposed technique is applied for proving that the chromatic index of a graph is equal to the chromatic index of its semi-core, the subgraph induced by the maximum degree vertices and their neighbors. Finally, we apply these decomposition tools to a classical result [A.J.W. Hilton, Z. Cheng, The chromatic index of a graph whose core has maximum degree 2, Discrete Math. 101 (1992) 135-147] that relates the chromatic index of a graph to its core, the subgraph induced by the maximum degree vertices.     

Chordal graphs and join spaces

A join space is an abstract model for partially ordered linear, spherical and projective geometries. A characterization for chordal graphs which are join spaces is given: a chordal graph is a join space if and only if it does not contain one of the two forbidden graphs as an induced subgraph.     

On the join of graphs and chromatic uniqueness

A graph is chromatically unique if it is uniquely determined by its chromatic polynomial. Let G be a chromatically unique graph and let K_m denote the complete graph on m vertices. This paper is mainly concerned with the chromaticity of K_m + G where + denotes the join of graphs. Also, it is shown that a large family of connected vertextransitive graphs that are not chromatically unique can be obtained by taking the join of some vertex-transitive graphs. © 1995 John Wiley & Sons, Inc.     

A class of antimagic join graphs

A labeling f of a graph G is a bijection from its edge set E(G) to the set {1, 2, . . . , |E(G)|}, which is antimagic if for any distinct vertices x and y, the sum of the labels on edges incident to x is different from the sum of the labels on edges incident to y. A graph G is antimagic if G has an f which is antimagic. Hartsfield and Ringel conjectured in 1990 that every connected graph other than K 2 is antimagic. In this paper, we show that if G 1 is an n-vertex graph with minimum degree at least r, and G 2 is an m-vertex graph with maximum degree at most 2r-1 (m ≥ n), then G1 ∨ G2 is antimagic.     

Ear decompositions of join covered graphs

Join covered graphs are ±1-weighted graphs, without negative circuits, in which every edge lies in a zero-weight circuit. Join covered graphs are a natural generalization of matching covered graphs. Many important properties of matching covered graphs have been generalized to join covered graphs. In this paper, we generalize Lovász and Plummerʼs ear decomposition theorem of matching covered graphs to join covered graphs.     

联图的圈基

MacLane于1937年给出了圈基方面的重要定理: 图G是平面图, 当且仅当图G有2-重基. 连通图G_1和G_2的联图G_1\vee G_2指的是在它们的不交并G_1\bigcup G_2上添加边集(u,v)|u\in V(G_1), v\in V(G_2). 对G_1和G_2的联图G_1\vee G_2的圈基重数进行了研究, 得到了一个上界, 改进了Zare的结果. 并在此基础之上, 进一步得到特殊联图C_m\vee C_n的圈基重数的一个上界.     

The inertia set of the join of graphs

Let G=(V,E) be a graph with V={1,2,…,n}. Denote by S(G) the set of all real symmetric n×n matrices A=[a_i,j] with a_i,j≠0, i≠j if and only if ij is an edge of G. Denote by I^↗(G) the set of all pairs (p,q) of natural numbers such that there exists a matrix A∈S(G) with at most p positive and q negative eigenvalues. We show that if G is the join of G₁ and G₂, then I^↗(G)?{(1,1)}=I^↗(G₁∨K₁)∩I^↗(G₂∨K₁)?{(1,1)}. Further, we show that if G is a graph with s isolated vertices, then , where denotes the graph obtained from G be removing all isolated vertices, and we give a combinatorial characterization of graphs G with (1,1)∈I^↗(G). We use these results to determine I^↗(G) for every complete multipartite graph G.     

On spanning connected graphs

A k-containerC(u,v) of G between u and v is a set of k internally disjoint paths between u and v. A k-container C(u,v) of G is a k^*-container if the set of the vertices of all the paths in C(u,v) contains all the vertices of G. A graph G is k^*-connected if there exists a k^*-container between any two distinct vertices. Therefore, a graph is 1^*-connected (respectively, 2^*-connected) if and only if it is hamiltonian connected (respectively, hamiltonian). In this paper, a classical theorem of Ore, providing sufficient conditional for a graph to be hamiltonian (respectively, hamiltonian connected), is generalized to k^*-connected graphs.     

On path connected graphs

Given n and i, n > 2, 2 ≤ i ≤ n ? 1, the smallest size of an n-graph without endvertices is obtained, which ensures a path of length i between any two vertices of the graph.