TY - JOUR

T1 - Properly Colored Subgraphs and Rainbow Subgraphs in Edge-Colorings with Local Constraints

AU - Alon, Noga

AU - Jiang, Tao

AU - Miller, Zevi

AU - Pritikin, Dan

PY - 2003/12

Y1 - 2003/12

N2 - We consider a canonical Ramsey type problem. An edge-coloring of a graph is called m-good if each color appears at most m times at each vertex. Fixing a graph G and a positive integer m, let f(m, G) denote the smallest n such that every m-good edge-coloring of K n yields a properly edge-colored copy of G, and let g(m, G) denote the smallest n such that every m-good edge-coloring of K n yields a rainbow copy of G. We give bounds on f(m, G) and g(m, G). For complete graphs G = K t we have c 1 mt 2/ln t ≤ f(m, K t) ≤ c 2mt 2, and c′ 1mt 3/ln t ≤ g(m, K t) ≤ c′ 2mt 3/ln t, where c 1, c 2, c′ 1, c′ 2 are absolute constants. We also give bounds on f(m, G) and g(m, G) for general graphs G in terms of degrees in G. In particular, we show that for fixed m and d, and all sufficiently large n compared to m and d, f(m, G) = n for all graphs G with n vertices and maximum degree at most d.

AB - We consider a canonical Ramsey type problem. An edge-coloring of a graph is called m-good if each color appears at most m times at each vertex. Fixing a graph G and a positive integer m, let f(m, G) denote the smallest n such that every m-good edge-coloring of K n yields a properly edge-colored copy of G, and let g(m, G) denote the smallest n such that every m-good edge-coloring of K n yields a rainbow copy of G. We give bounds on f(m, G) and g(m, G). For complete graphs G = K t we have c 1 mt 2/ln t ≤ f(m, K t) ≤ c 2mt 2, and c′ 1mt 3/ln t ≤ g(m, K t) ≤ c′ 2mt 3/ln t, where c 1, c 2, c′ 1, c′ 2 are absolute constants. We also give bounds on f(m, G) and g(m, G) for general graphs G in terms of degrees in G. In particular, we show that for fixed m and d, and all sufficiently large n compared to m and d, f(m, G) = n for all graphs G with n vertices and maximum degree at most d.

UR - http://www.scopus.com/inward/record.url?scp=0344287271&partnerID=8YFLogxK

U2 - 10.1002/rsa.10102

DO - 10.1002/rsa.10102

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AN - SCOPUS:0344287271

SN - 1042-9832

VL - 23

SP - 409

EP - 433

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

IS - 4

ER -