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Partitioning infinite trees
Author(s) -
Horak Peter,
Heinrich Katherine
Publication year - 2000
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/1097-0118(200006)34:2<113::aid-jgt1>3.0.co;2-0
Subject(s) - combinatorics , mathematics , monochromatic color , degree (music) , graph , discrete mathematics , upper and lower bounds , tree (set theory) , mathematical analysis , physics , botany , acoustics , biology
A graph G is bisectable if its edges can be colored by two colors so that the resulting monochromatic subgraphs are isomorphic. We show that any infinite tree of maximum degree Δ with infinitely many vertices of degree at least Δ −1 is bisectable as is any infinite tree of maximum degree Δ ≤ 4. Further, it is proved that every infinite tree T of finite maximum degree contains a finite subset E of its edges so that the graph T − E is bisectable. To measure how “far” a graph G is from being bisectable, we define c ( G ) to be the smallest number k > 1 so that there is a coloring of the edges of G by k colors with the property that any two monochromatic subgraphs are isomorphic. An upper bound on c ( G ), which is in a sense best possible, is presented. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 113–127, 2000