We propose a new self-stabilizing 1-maximal matching algorithm which is silent and works for any anonymous networks without a cycle of length of a multiple of 3 under a central unfair daemon. The 1-maximal matching is a 2/3-approximation to the maximum matching, and expected to get more matching pairs than a maximal matching, which only guarantees a 1/2-approximation. The time complexity of the proposed algorithm is O ( e ) moves, which is O ( n ) moves if we restrict the topology to trees or rings whose length is not a multiple of 3, where n and e be the numbers of nodes and edges in a graph, respectively. The best existing result for 1-maximal matching for anonymous networks is an algorithm of Goddard et al. which works for anonymous trees and anonymous rings whose length is not a multiple of 3 under a central daemon, and the time complexity is O ( n 4) moves. Therefore, the result in this paper is a significant improvement from the best existing results.
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