A face of an edge colored plane graph is called rainbow if all its edges receive distinct colors. The maximum number of colors used in an edge coloring of a connected plane graph G with no rainbow face is called the edge-rainbowness of G. In this paper we prove that the edge-rainbowness of G equals to the maximum number of edges of a connected bridge face factor H of G, where a bridge face fact...

Numerous networks as, for example, road networks, electrical networks and communication networks can be modeled by a graph. Many attempts have been made to determine how well such a network is "connected" or stated differently how much effort is required to break down communication in the system between at least some nodes. Two well-known measures that indicate how "reliable" a graph is are the...

Let G be a graph and T a set of vertices. A T -path in G is a path that begins and ends in T , and none of its internal vertices are contained in T . We define a T -path covering to be a union of vertex-disjoint T paths spanning all of T . Concentrating on graphs that are tough (the removal of any nonempty set X of vertices yields at most |X| components), we completely characterize the edges th...

The problem in which the object is to add a minimum weight set of edges to a graph so as to satisfy a given condition is called the augmentation problem. This problem has a wide variety [3-5, 10, 11, 14-201. If such a given condition is concerned with the vertexor edge-connectivity of a graph then the problem is referred to as the (vertexor edge-) connectivity augmentation problem. Frank and Ch...

The generalized k-connectivity κk(G) of a graph G was introduced by Chartrand et al. in 1984. It is natural to introduce the concept of generalized k-edge-connectivity, λk(G). For general k, the generalized k-edgeconnectivity of a complete graph is obtained. For k ≥ 3, tight upper and lower bounds of κk(G) and λk(G) are given for a connected graph G of order n, namely, 1 ≤ κk(G) ≤ n− k2 and 1 ≤...

An edge cut W of a connected graph G is a k-restricted edge cut if G−W is disconnected, and every component of G −W has at least k vertices. The k-restricted edge connectivity is defined as the minimum cardinality over all k-restricted edge cuts. A permutation graph is obtained by taking two disjoint copies of a graph and adding a perfect matching between the two copies. The k-restricted edge c...

We give the first O(t log t) time algorithm for processing a given sequence of t edge updates and 3-vertex/edge connectivity queries in an undirected unweighted graph. Our approach builds upon a method by Eppstein [Epp94] that reduces graphs to smaller equivalents on a set of key vertices. It also leads to algorithms for offline 2-vertex/edge connectivity whose performances match those from Kac...

For a connected graph G = (V, E), an edge set S ⊆ E is a k-restricted edge cut if G−S is disconnected and every component of G− S has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is defined as the cardinality of a minimum k-restricted edge cut. Let ξk(G) = min{|[X, X ]| : |X | = k,G[X ] is connected}. G is λk -optimal if λk(G) = ξk(G). Moreover, G is super-λk ...

Fault tolerant measures have played an important role in the reliability of an interconnection network. Edge connectivity, restricted-edge-connectivity, extra-edge-connectivity and super-edge-connectivity of many well-known interconnection networks have been explored. In this paper, we study the 2-extra-edge connectivity of a special class of graphs G(G0,G1; M) proposed by Chen et al. [Appl. Ma...