Abstract
AbstractLet G be a loopless finite multigraph. For each vertex x of G, denote its degree and multiplicity by d(x) and μ(x) respectively. DefineØ(x) = the least even integer ≥ μ(x), if d(x) is even, the least odd integer ≥ μ(x), if d(x) is odd.In this paper it is shown that every multigraph G admits a faithful path decomposition—a partition P of the edges of G into simple paths such that every vertex x of G is an end of exactly Ø(x) paths in P. This result generalizes Lovász's path decomposition theorem, Li's perfect path double cover theorem (conjectured by Bondy), and a result of Fan concerning path covers of weighted graphs. It also implies an upper bound on the number of paths in a minimum path decomposition of a multigraph, which motivates a generalization of Gallai's path decomposition conjecture. © 1995 John Wiley & Sons, Inc.
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