We present a simple depth-rst search strategy for exploring (constructing) an unknown strongly connected graph G with m edges and n vertices by traversing at most min(mn; dn 2 + m) edges. Here, d is the minimum number of edges needed to add to G to make it Eulerian. This parameter d is known as the deeciency of a graph and was introduced by Kutten Kut88]. It was conjectured that graphs with low...

In this paper, we study fractional multiflows in undirected graphs. A fractional multiflow in a graph G with a node subset T, called terminals, is a collection of weighted paths with ends in T such that the total weights of paths traversing each edge does not exceed 1. Well-known fractional path packing problem consists of maximizing the total weight of paths with ends in a subset S of TxT over...

We prove that every Eulerian orientation of Km,n contains 1 4+ √ 8 mn(1 − o(1)) arc-disjoint directed 4-cycles, improving earlier lower bounds. Combined with a probabilistic argument, this result is used to prove that every regular tournament with n vertices contains 1 8+ √ 32 n2(1−o(1)) arc-disjoint directed 4-cycles. The result is also used to provide an upper bound for the distance between t...

It is based on well-known network models Euler graph, Erdös and Renyi random graph, Watts-Strogatz small-world model and Barabási-Albert scale-free networks, and combined the unified hybrid network theoretical frame. One kind of network complexity pyramid with universality and diversity is constructed, described and reviewed. It is found that most unweighted and weighted models of network scien...

P.D. Seymour proved that the cut criterion is necessary and sufficient for the solvability of the edgedisjoint paths problem when the union of the supply graph and the demand graph is planar and Eulerian. When only planarity is required, M. Middendorf and F. Pfeiffer proved the problem to be NP-complete. For this case E. Korach and M. Penn proved that the cut criterion is sufficient for the exi...

Upper bounds on the maximum number of minimal codewords in a binary code follow from the theory of matroids. Random coding provide lower bounds. In this paper we compare these bounds with analogous bounds for the cycle code of graphs. This problem (in the graphic case) was considered in 1981 by Entringer and Slater who asked if a connected graph with p vertices and q edges can have only slightl...

P.D. Seymour proved that the cut criterion is necessary and suucient for the solvability of the edge-disjoint paths problem when the union of the supply graph and the demand graph is planar and Eulerian. When only planarity is required, M. Middendorf and F. Pfeiier proved the problem to be NP-complete. For this case E. Korach and M. Penn proved that the cut criterion is suucient for the existen...

This thesis considers a novel problem of obtaining global channel state information (CSI) at every node in an ad-hoc wireless network. A class of protocols for dissemination and estimation are developed which attempt to minimize the staleness of the estimates throughout the network. This thesis also provides an optimal protocol for CSI dissemination in networks with complete graph topology and ...

A classic theorem of Veblen states that a connected graph G has a cycle decomposition if and only if G is Eulerian. The number of odd cycles in a cycle decomposition of an Eulerian graph G is therefore even if and only if G has even size. It is conjectured that if the minimum number of odd cycles in a cycle decomposition of an Eulerian graph G with m edges is a and the maximum number of odd cyc...

In this paper, we develop a new multicasting model for wormhole-routed networks based on the concept of Euler path in graph theory. The model can support multiple multicasts freely from deadlock and can be applied to any network which is Eulerian or is Eulerian after some links being removed. In particular, we demonstrate the power of this model by showing its fault-tolerant capability in suppo...