The sequential join of combinatorial games.
For a nontrivial connected graph G, let c: V(G)→ N be a vertex coloring of G where adjacent vertices may be colored the same. For a vertex v of G, the neighborhood color set NC(v) is the set of colors of the neighbors of v. The coloring c is called a set coloring if NC(u) ≠ NC(v) for every pair u,v of adjacent vertices of G. The minimum number of colors required of such a coloring is called the set chromatic number χₛ(G) of G. The set chromatic numbers of some well-known classes of graphs are determined...
Let G be a graph with vertex set V(G) and edge set E(G). A signed matching is a function x: E(G) → -1,1 satisfying for every v ∈ V(G), where . The maximum of the values of , taken over all signed matchings x, is called the signed matching number and is denoted by β’₁(G). In this paper, we study the complexity of the maximum signed matching problem. We show that a maximum signed matching can be found in strongly polynomial-time. We present sharp upper and lower bounds on β’₁(G) for general graphs....
In this paper, we determine the graph with maximal signless Laplacian spectral radius among all connected graphs with fixed order and given number of cut vertices.
In this paper, the effects on the signless Laplacian spectral radius of a graph are studied when some operations, such as edge moving, edge subdividing, are applied to the graph. Moreover, the largest signless Laplacian spectral radius among the all unicyclic graphs with vertices and pendant vertices is identified. Furthermore, we determine the graphs with the largest Laplacian spectral radii among the all unicyclic graphs and bicyclic graphs with vertices and pendant vertices, respectively....
A 3-uniform hypergraph is called a minimum 3-tree, if for any 3-coloring of its vertex set there is a heterochromatic triple and the hypergraph has the minimum possible number of triples. There is a conjecture that the number of triples in such 3-tree is ⎡(n(n-2))/3⎤ for any number of vertices n. Here we give a proof of this conjecture for any n ≡ 0,1 mod 12.
We study random circle graphs which are generated by throwing n points (vertices) on the circle of unit circumference at random and joining them by an edge if the length of shorter arc between them is less than or equal to a given parameter d. We derive here some exact and asymptotic results on sizes (the numbers of vertices) of "typical" connected components for different ways of sampling them. By studying the joint distribution of the sizes of two components, we "go into" the structure of random...
The connectivity and measure theoretic properties of the skeleta of convex bodies in Euclidean space are discussed, together with some long standing problems and recent results.
Let be a finite group. The main supergraph is a graph with vertex set in which two vertices and are adjacent if and only if or . In this paper, we will show that if and only if . As a main consequence of our result we conclude that Thompson’s problem is true for the small Ree group .