Edge-Coloring Bipartite Multigraphs in O(E logD) Time

  title={Edge-Coloring Bipartite Multigraphs in O(E logD) Time},
  author={Richard J. Cole and Kirstin Ost and Stefan Schirra},
Let V, E, and D denote the cardinality of the vertex set, the cardinality of the edge set, and the maximum degree of a bipartite multigraph G. We show that a minimal edge-coloring of G can be computed in O(E logD time. This result follows from an algorithm for finding a matching in a regular bipartite graph in O(E) time. 

Topics from this paper

Generalized edge-colorings of weighted graphs
Let G be a graph with a positive integer weight ω(v) for each vertex v. One wishes to assign each edge e of G a positive integer f(e) as a color so that ω(v) ≤|f(e) − f(e′)| for any vertex v and an...
Another Simple Algorithm for Edge-Coloring Bipartite Graphs
  • Takashi Takabatake
  • Mathematics, Computer Science
  • IEICE Trans. Fundam. Electron. Commun. Comput. Sci.
  • 2005
A new edge-coloring algorithm for bipartite graphs that does not require elaborate data structures, which the best known O(mlogd) algorithm due to Cole--Ost--Schirra depends on is presented. Expand
A Self-Stabilizing ( Δ + 4 )-Edge-Coloring Algorithm for Planar Graphs in Anonymous Uniform Systems
This paper proposes a self-stabilizing edge coloring algorithm using (Δ + 4) colors for distributed systems of a planar graph topology, where Δ ≥ 5 is the maximum degree of the graph. The algorithmExpand
A self-stabilizing (Delta+4)-edge-coloring algorithm for planar graphs in anonymous uniform systems
A self-stabilizing edge-coloring algorithm using (@D+4) colors for distributed systems of a planar graph topology, where @D>=5 is the maximum degree of the graph. Expand
Subset matching and edge coloring in bipartite graphs
A necessary and sufficient condition always holds when the subset is composed of the vertices with maximum degree, which leads to a simple algorithm that finds an optimal edge coloring in bipartite graphs with no need to transform the bipartITE graph into a regular one. Expand
Maximum Matching in Regular and Almost Regular Graphs
  • R. Yuster
  • Mathematics, Computer Science
  • Algorithmica
  • 2012
An O(n2logn)-time algorithm that finds a maximum matching in a regular graph with n vertices that is faster than applying the fastest known general matching algorithm that runs in O(\sqrt{n}m)$-time for graphs with m edges. Expand
An asymptotic approximation scheme for multigraph edge coloring
This work gives polynomial time algorithms for approximate edge coloring of multigraphs, i.e., parallel edges are allowed, and achieves arbitrarily good approximation factors at the cost of slightly larger additive terms. Expand
Total Colorings Of Degenerate Graphs
It is proved that an s-degenerate graph G has a total coloring with Δ+1 colors if the maximum degree Δ of G is sufficiently large, say Δ≥4s+3. Expand
Linear algorithm for selecting an almost regular spanning subgraph in an almost regular graph
An algorithm for selecting an almost d′-regular spanning subgraph in optimal linear time is given. Expand
An Algorithm for Computing Edge Colorings on Regular Bipartite Multigraphs
In this paper, we consider the problem of finding an edge coloring of a d-regular bipartite multigraph with 2n vertices and m = nd edges. The best known deterministic algorithm (by Cole, Ost, andExpand


On Edge Coloring Bipartite Graphs
The present paper shows how to find a minimal edge coloring of a bipartite graph with E edges and V vertices in time $O(E\log V)$.
Edge-Coloring Algorithms
Various upper bounds on the minimum number of colors required to edge-color graphs are reviewed, and efficient algorithms toEdge-coloring graphs with a number of Colors not exceeding the upper bounds are presented. Expand
Edge-Coloring Bipartite Graphs
Given a bipartite graph G with n nodes, m edges, and maximum degree ?, we find an edge-coloring for G using ? colors in time T+O(mlog?), where T is the time needed to find a perfect matching in aExpand
Algorithms for edge coloring bipartite graphs
Coloring algorithms are presented that use time O(min(¦E¦ &Dgr; log n) and space O(n&DGr;) and find maximum matchings on regular (or semi-regular) bipartite graphs. Expand
An n5/2 Algorithm for Maximum Matchings in Bipartite Graphs
The present paper shows how to construct a maximum matching in a bipartite graph with n vertices and m edges in a number of computation steps proportional to $(m + n)\sqrt n $.
The NP-Completeness of Edge-Coloring
  • I. Holyer
  • Mathematics, Computer Science
  • SIAM J. Comput.
  • 1981
It is shown that it is NP-complete to determine the chromatic index of an arbitrary graph, even for cubic graphs. Expand
A fast parallel algorithm for routing in permutation networks
An algorithm is given for routing in permutation networks-that is, for computing the switch settings that implement a given permutation. The algorithm takes serial time <i>O</i>(<i>n</i>(logExpand
Two problems in graph theory
Firstly, two approaches to the problem of finding a minimal edge coloring of a bipartite graph are presented. These yield algorithms with running times of 0(E log D + V log V log('3) D), and 0(E logExpand