TY - JOUR

T1 - An effcient algorithm for the nearly equitable edge coloring problem

AU - Xie, Xuzhen

AU - Yagiura, Mutsunori

AU - Ono, Takao

AU - Hirata, Tomio

AU - Zwick, Uri

PY - 2008

Y1 - 2008

N2 - An edge coloring of a multigraph is nearly equitable if, among the edges incident to each vertex, the numbers of edges colored with any two colors differ by at most two. It has been proved that the problem of finding a nearly equitable edge coloring can be solved in O(m2=k) time, where m and k are the numbers of edges and given colors, respectively. In this pa- per, we present a recursive algorithm that runs in O (mnlog (m=(kn) + 1)) time, where n is the number of vertices. This algorithm improves the best- known worst-case time complexity. When k = O(1), the time complexity of all known algorithms is O(m2), which implies that this time complex- ity remains to be the best for more than twenty years since 1982 when Hilton and de Werra gave a constructive proof for the existence of a nearly equitable edge coloring for any multigraph. Our result is the first that im- proves this time complexity when m=n grows to infinity; e.g., m = nθ for an arbitrary constant θ > 1.

AB - An edge coloring of a multigraph is nearly equitable if, among the edges incident to each vertex, the numbers of edges colored with any two colors differ by at most two. It has been proved that the problem of finding a nearly equitable edge coloring can be solved in O(m2=k) time, where m and k are the numbers of edges and given colors, respectively. In this pa- per, we present a recursive algorithm that runs in O (mnlog (m=(kn) + 1)) time, where n is the number of vertices. This algorithm improves the best- known worst-case time complexity. When k = O(1), the time complexity of all known algorithms is O(m2), which implies that this time complex- ity remains to be the best for more than twenty years since 1982 when Hilton and de Werra gave a constructive proof for the existence of a nearly equitable edge coloring for any multigraph. Our result is the first that im- proves this time complexity when m=n grows to infinity; e.g., m = nθ for an arbitrary constant θ > 1.

UR - http://www.scopus.com/inward/record.url?scp=76249115237&partnerID=8YFLogxK

U2 - 10.7155/jgaa.00171

DO - 10.7155/jgaa.00171

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AN - SCOPUS:76249115237

SN - 1526-1719

VL - 12

SP - 383

EP - 399

JO - Journal of Graph Algorithms and Applications

JF - Journal of Graph Algorithms and Applications

IS - 4

ER -