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A FRACTIONAL ANALOGUE OF BROOKS' THEOREM.
- Source :
-
SIAM Journal on Discrete Mathematics . 2012, Vol. 26 Issue 2, p452-471. 20p. 20 Diagrams. - Publication Year :
- 2012
-
Abstract
- Let Δ (G) be the maximum degree of a graph G. Brooks' theorem states that the only connected graphs with chromatic number X(G) = Δ (G) + 1 are complete graphs and odd cycles. We prove a fractional analogue of Brooks' theorem in this paper. Namely, we classify all connected graphs G such that the fractional chromatic number Xf(G) is at least Δ (G). These graphs are complete graphs, odd cycles, C²8 C5 ⊠ K2, and graphs whose clique number ω(G) equals the maximum degree Δ (G). Among the two sporadic graphs, the graph C²8 is the square graph of cycle C8, while the other graph C5 ⊠ K2 is the strong product of C5 and K2. In fact, we prove a stronger result: If a connected graph G with δ (G) ≥ 4 is not one of the graphs listed above, then we have Xf(G)≤ Δ(G) -- 2/67. [ABSTRACT FROM AUTHOR]
- Subjects :
- *GRAPH theory
*PROBABILITY theory
*MATHEMATICS
*FRACTIONS
*REAL numbers
Subjects
Details
- Language :
- English
- ISSN :
- 08954801
- Volume :
- 26
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- SIAM Journal on Discrete Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 80535679
- Full Text :
- https://doi.org/10.1137/110827879