It is well known that the chromatic index of any bipartite multigraph G with n vertices and m edges is equal to its maximum degree k. The best algorithm currently known for finding a proper k-edge-coloring of such a multigraph runs in time O(m log k), see [2], or the forthcoming book [6], and applies rather elaborate data structures together with the basic approach of [5]. Another algorithm, of...

For a loopless multigraph G, the fractional arboricity Arb(G) is the maximum of |E(H)| |V (H)|−1 over all subgraphs H with at least two vertices. Generalizing the NashWilliams Arboricity Theorem, the Nine Dragon Tree Conjecture asserts that if Arb(G) ≤ k + d k+d+1 , then G decomposes into k + 1 forests with one having maximum degree at most d. The conjecture was previously proved for d = k+1 an...

There are 22521 nonisomorphic 2-(9,3,3) designs of which 9218 are decomposable and 395 resolvable. Computational methods used to find and analyse these designs are discussed. All cubic multigraphs on 8 vertices are displayed and their role in the generation process is outlined. Statistics are presented concerning neighborhood graphs, multiple blocks, parallel classes, subdesigns, group orders a...

C. A. Rodger, M. A. Tiemeyer∗, Auburn University Let K = K(a, p;λ1, λ2) be the multigraph with: the number of vertices in each part equal to a; the number of parts equal to p; the number of edges joining any two vertices of the same part equal to λ1; and the number of edges joining any two vertices of different parts equal to λ2. This graph was of interest to Bose and Shimamoto in their study o...

P. Erdős and T. Gallai gave necessary and sufficient conditions for a sequence of non-negative integers to be graphic. Here,their result is generalized to multigraphs with a specified multiplicity. This both generalizes and provides a new proof of a result in the literature by Chungphaisan [2].

We present an unsupervised model for coreference resolution that casts the problem as a clustering task in a directed labeled weighted multigraph. The model outperforms most systems participating in the English track of the CoNLL’12 shared task.

An edge coloring of a graph G is a process of assigning colors to the adjacent edges so that the adjacent edges represents the different colors. In this paper, an algorithm is proposed to find the perfect color matching of the regular bipartite multigraph with low time complexity. For that, the proposed algorithm is divided into two procedures. In the first procedure, the possible circuits and ...

We discuss the nearly equitable edge coloring problem on a multigraph and propose an efficient algorithm for solving the problem, which has a better time complexity than the previous algorithms. The coloring computed by our algorithm satisfies additional balanced conditions on the number of edges used in each color class, where conditions are imposed on the balance among all edges in the multig...

Let G = (V,E) be a multigraph (it has multiple edges, but no loops). We call G maximally edge-connected if λ(G) = δ(G), and G super edge-connected if every minimum edge-cut is a set of edges incident with some vertex. The restricted edgeconnectivity λ′(G) of G is the minimum number of edges whose removal disconnects G into non-trivial components. If λ′(G) achieves the upper bound of restricted ...

This paper exploits the remarkable new method of Galvin (J. Combin. Theory Ser. B 63 (1995), 153 158), who proved that the list edge chromatic number /$list(G) of a bipartite multigraph G equals its edge chromatic number /$(G). It is now proved here that if every edge e=uw of a bipartite multigraph G is assigned a list of at least max[d(u), d(w)] colours, then G can be edge-coloured with each e...