A {1, 3, · · · , 2n−1}-factor of a graph G is defined to be a spanning subgraph of G, each degree of whose vertices is one of {1, 3, · · · , 2n− 1}, where n is a positive integer. In this paper, we give a sufficient condition for a graph to have a {1, 3, · · · , 2n− 1}-factor.

If we fix a spanning subgraph H of a graph G, we can define a chromatic number of H with respect to G and we show that it coincides with the chromatic number of a double covering of G with co-support H. We also find a few estimations for the chromatic numbers of H with respect to G.

We prove a general result on graph factors modulo k. A special case says that, for each natural number k, every (12k − 7)-edge-connected graph with an even number of vertices contains a spanning subgraph in which each vertex has degree congruent to k modulo 2k.

We present a linear time algorithm exactly solving the 2-edge connected spanning subgraph (2-ECSS) problem in a graph of bounded treewidth. Using this with Klein’s diameter reduction technique [15], we find a linear time PTAS for the problem in unweighted planar graphs, and the first PTAS for the problem in weighted planar graphs.

A linear time -approximation algorithm is presented for the NP-hard problem of finding a minimum strongly-connected spanning subgraph. It is based on cycle contraction that was first introduced by Khuller, Raghavachari and Young (1995). We improve their result by contracting special cycles and utilizing a more efficient data structure.

5 We give an optimal degree condition for a tripartite graph to have 6 a spanning subgraph consisting of complete graphs of order 3. This 7 result is used to give an upper bound of 2∆ for the strong chromatic 8 number of n vertex graphs with ∆ ≥ n/6. 9

Let f : X −→ N be an integer function. An f -factor is a spanning subgraph of a graph G = (X,E) whose vertices have degrees defined by f . In this paper, we prove a sufficient condition for the existence of a f -factor which involves the stability number, the minimun degree of G or the connectivity of the graph.

In this paper, we study the minimum size 2-edge connected spanning subgraph problem (henceforth 2EC) and show that every 3-edge connected cubic graphG = (V,E), with n = |V | allows a 2EC solution for G of size at most 7n 6 , which improves upon Boyd, Iwata and Takazawa’s guarantee of 6n 5 .

Let G be a connected n{order graph and suppose that n = n1 + + nk where ni 2 are integers. The following will be proved : If G has minimum degree at least 1 2n1 + + 12nk then G has a spanning subgraph which consists of paths of orders n1; : : : ; nk.

We survey results on online versions of the standard network optimization problems, including the minimum spanning tree problem, the minimum Steiner tree problem, the weighted and unweighted matching problems, and the traveling salesman problem. The goal in these problems is to maintain, with minimal changes, a low cost subgraph of some type in a dynamically changing network.