We give asymptotically sharp upper bounds for the maximum diameter and radius of (i) a connected graph, {ii) a connected trangle-free graph, (iii) a connected C,-free graph with n vertices and with minimum degree 6, where n tends to infinity. Some conjectures for J&-free graphs are also stated. Let G be a connected graph with vertex set V(G) and edge set E(G). For any XJJ E V(G) let dG(x,y) den...

Let X = (V, E) be a finite simple connected graph with n vertices and m edges. A configuration is an assignment of one of two colors, black or white, to each edge of X. A move applied to a configuration is to select a black edge 2 ∈ E and change the colors of all adjacent edges of 2. Given an initial configuration and a final configuration, try to find a sequence of moves that transforms the in...

A circle in a graph G is a homeomorphic image of the unit circle in the Freudenthal compactification of G, a topological space formed from G and the ends of G. Bruhn conjectured that every locally finite 4-connected planar graph G admits a Hamilton circle, a circle containing all points in the Freudenthal compactification of G that are vertices and ends of G. We prove this conjecture for graphs...

For a positive integer k ≥ 4, a graph G is called k-ordered, if for any ordered set of k distinct vertices of G, G has a cycle that contains all the vertices in the designated order. Goddard [3] showed that every 4-connected triangulation of the plane is 4-ordered. In this paper, we improve this result; every 4-connected triangulation of any surface is 4-ordered. Our proof is much shorter than ...

For c 2 and k minfc; 3g, guaranteed upper bounds on the length of a shortest cycle through k prescribed vertices of a c-connected graph are proved. Analogous results on planar graphs are presented, too.

Given a graph G = (V,E) and a set of terminal vertices T we say that a superset S of T is T -connecting if S induces a connected graph, and S is minimal if no strict subset of S is T -connecting. In this paper we prove that there are at most (|V \T | |T |−2 ) · 3 |V \T | 3 minimal T -connecting sets when |T | ≤ n/3 and that these can be enumerated within a polynomial factor of this bound. This ...

We adapt the classical 3-decomposition of any 2-connected graph to the case of simple graphs (no loops or multiple edges). By analogy with the block-cutpoint tree of a connected graph, we deduce from this decomposition a bicolored tree tc(g) associated with any 2-connected graph g, whose white vertices are the 3-components of g (3-connected components or polygons) and whose black vertices are b...

We introduce one of many classes of problems which can be defined in terms of 3-valued functions on the vertices of a graph G = (V, E) of the form f : V + { 1, 0, l}. Such a fknction is said to be a minus dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every t’ E V, ,f(N[r~])> 1, where N[a] consists of 1: and every vertex adjacent...

Let G be a simple connected graph. The terminal distance matrix of G is the distance matrix between all pendant vertices of G. In this paper, we introduce some general transformations that increase the terminal distance spectral radius of a connected graph and characterize the extremal trees with respect to the terminal distance spectral radius among all trees with a fixed number of pendant ver...

The old well-known result of Chartrand, Kaugars and Lick [1] says that every k-connected graph G with minimum degree at least 3k/2 has a vertex v such that G− v is still k-connected. In this paper, we consider a generalization of the above result. We prove the following result: Suppose G is a k-connected graph with minimum degree at least 3k/2 + 2. Then G has an edge e such that G− V (e) is sti...