The bounded chromatic number for graphs of genus g

• Published in: Journal of combinatorial theory. Series B Vol. 56; no. 2; pp. 183 - 196
• Main Authors: Berman, Kenneth A, Paul, Jerome L
• Format: Journal Article
• Language: English
• Published: Duluth, MN Elsevier Inc 01.11.1992; Academic Press
• Subjects: Combinatorics; Combinatorics. Ordered structures; Exact sciences and technology; Graph theory; Mathematics; Sciences and techniques of general use; Edge(graph); Chromatic number; Finite graph; Extremum problem
• ISSN: 0095-8956; 1096-0902
• DOI: 10.1016/0095-8956(92)90017-R

For G a collection of finite graphs, the bounded chromatic number χ B( G ) is the smallest number of colors c for which there exists an integer N such that every graph G∈ G can be vertex c-colored without forcing more than N monochromatic edges. The bounded (simple) path chromatic number χP( G ) ( χSP( G )) is the smallest number of colors c for which there exists an integer N such that every graph G∈ G can be c-colored without forcing a monochromatic (simple) path of length more than N. For the set S g of all graphs of genus g it is known that 4≤ χB( S g)≤6, and χSP( S g)=4. In this paper we show that χB( S g)≤5, and χP( S g)=4. For g≥1, let μ 5( g) ( π 4( g)) denote the smallest integer x such that every graph G∈ S g can be 5-colored without forcing more than x monochromatic edges (4-colored without forcing a monochromatic path of length more than x). We also show that 2 g ≤ μ 5( g) ≤ 74 g − 36 and ( 3 8 )g − ( 1 2 ) 3g + 3 32 ≤ π 4(g) ≤ 224g − 106 .