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On sufficient conditions for rainbow cycles in edge-colored graphs.
- Source :
-
Discrete Mathematics . Jul2019, Vol. 342 Issue 7, p1956-1965. 10p. - Publication Year :
- 2019
-
Abstract
- Let G be an edge-colored graph. We use e (G) and c (G) to denote the number of edges of G and the number of colors appearing on E (G) , respectively. For a vertex v ∈ V (G) , the color neighborhood of v is defined as the set of colors assigned to the edges incident to v. A subgraph of G is rainbow if all of its edges are assigned with distinct colors. The well-known Mantel's theorem states that a graph G on n vertices contains a triangle if e (G) ≥ ⌊ n 2 4 ⌋ + 1. Rademacher (1941) [8] showed that G contains at least ⌊ n 2 ⌋ triangles under the same condition. Li et al. (2014) proved a rainbow version of Mantel's theorem: An edge-colored graph G has a rainbow triangle if e (G) + c (G) ≥ n (n + 1) ∕ 2. In this paper, we first characterize all graphs G satisfying e (G) + c (G) ≥ n (n + 1) ∕ 2 − 1 but containing no rainbow triangles. Motivated by Rademacher's theorem, we then characterize all graphs G which satisfy e (G) + c (G) ≥ n (n + 1) ∕ 2 but contain only one rainbow triangle. We further obtain two results on color neighborhood conditions for the existence of rainbow short cycles. Our results improve a previous theorem due to Broersma et al. (2005). Moreover, we provide a sufficient condition in terms of color neighborhood for the existence of a specified number of vertex-disjoint rainbow cycles. [ABSTRACT FROM AUTHOR]
- Subjects :
- *RAINBOWS
*CYCLES
*TRIANGLES
*NEIGHBORHOODS
*COLORS
*GEOMETRIC vertices
Subjects
Details
- Language :
- English
- ISSN :
- 0012365X
- Volume :
- 342
- Issue :
- 7
- Database :
- Academic Search Index
- Journal :
- Discrete Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 136462628
- Full Text :
- https://doi.org/10.1016/j.disc.2019.03.012