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- W292709859 abstract "1 Multicolored parallelisms in complete graphs Throughout this paper Ks denotes the complete graph on s vertices. We color the edges of K2n with 2n−1 colors by assigning one color to each edge. Basic terminology and notation on graph theory is found in Berge (1971). The coloration is proper if whenever two edges that have one vertex in common carry different colors. A spanning tree is called multicolored if no two of its edges have the same color. Two trees are edge disjoint if they do not share common edges. Two graphs with colored edges are isomorphic if there exists a bijection σ between the sets of vertices and a bijection η between the sets of colors such that (i, j) is an edge of color c if and only if (σ(i),σ( j)) is an edge of color η(c). We investigate the possibility of producing a proper edge-coloration of K2n such that its edges can be partitioned into edge disjoint isomorphic multicolored spanning trees. [By isomorphic multicolored spanning trees we understand a set of spanning trees, each of which is multicolored, any two spanning trees of the set being isomorphic as uncolored spanning trees.] When this is possible to accomplish we obtain what we call a multicolored tree parallelism for K2n. When no coloring is involved, it is well-known, and a classical result of Euler, that the edges of K2n can be partitioned into isomorphic spanning trees (paths, for example). Each of these spanning trees can easily be made multicolored, but the resulting edge coloration of K2n usually fails to be proper. Indeed, there exists a proper coloration of K8 that does not admit a multicolored path; see Buliga (2002), [8]. By an inductive construction we demonstrate that a partition of the edges of Km into edge-disjoint isomorphic multicolored spanning trees that induce a proper coloration of Km is possible whenever m (> 4) is a power of two, or five times a power of two." @default.
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- W292709859 date "2002-01-01" @default.
- W292709859 modified "2023-09-27" @default.
- W292709859 title "Multicolored parallelisms of isomorphic spanning trees" @default.
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