TY - JOUR

T1 - Breaking the rhythm on graphs

AU - Alon, Noga

AU - Grytczuk, Jarosław

N1 - Funding Information:
The first author was supported by an ISF grant and by a USA-Israeli BSF grant. The second author was supported by Grant KBN 1P03A 017 27.

PY - 2008/4/28

Y1 - 2008/4/28

N2 - We study graph colorings avoiding periodic sequences with large number of blocks on paths. The main problem is to decide, for a given class of graphs F, if there are absolute constants t, k such that any graph from the class has a t-coloring with no k identical blocks in a row appearing on a path. The minimum t for which there is some k with this property is called the rhythm threshold of F, denoted by t (F). For instance, we show that the rhythm threshold of graphs of maximum degree at most d is between (d + 1) / 2 and d + 1. We give several general conditions for finiteness of t (F), as well as some connections to existing chromatic parameters. The question whether the rhythm threshold is finite for planar graphs remains open.

AB - We study graph colorings avoiding periodic sequences with large number of blocks on paths. The main problem is to decide, for a given class of graphs F, if there are absolute constants t, k such that any graph from the class has a t-coloring with no k identical blocks in a row appearing on a path. The minimum t for which there is some k with this property is called the rhythm threshold of F, denoted by t (F). For instance, we show that the rhythm threshold of graphs of maximum degree at most d is between (d + 1) / 2 and d + 1. We give several general conditions for finiteness of t (F), as well as some connections to existing chromatic parameters. The question whether the rhythm threshold is finite for planar graphs remains open.

KW - Graph coloring

KW - Random graph

KW - Rhythm threshold

KW - Thue sequence

UR - http://www.scopus.com/inward/record.url?scp=38849190648&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2007.07.063

DO - 10.1016/j.disc.2007.07.063

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AN - SCOPUS:38849190648

SN - 0012-365X

VL - 308

SP - 1375

EP - 1380

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 8

ER -