Affiliation:
1. Center for Combinatorics and LPMC Nankai University Tianjin China
2. College of Computer Science Nankai University Tianjin China
3. School of Mathematics and Statistics Northwestern Polytechnical University Xi'an Shaanxi China
4. Xi'an‐Budapest Joint Research Center for Combinatorics Northwestern Polytechnical University Xi'an Shaanxi China
Abstract
AbstractLet be an edge‐colored graph on vertices. The minimum color degree of , denoted by , is defined as the minimum number of colors assigned to the edges incident to a vertex in . In 2013, Li proved that an edge‐colored graph on vertices contains a rainbow triangle if . In this paper, we obtain several estimates on the number of rainbow triangles through one given vertex in . As a consequence, we prove counting results for rainbow triangles in edge‐colored graphs. One main theorem states that the number of rainbow triangles in is at least , which is best possible by considering the rainbow ‐partite Turán graph, where its order is divisible by . This means that there are rainbow triangles in if , and rainbow triangles in if when . Both results are tight in the sense of the order of the magnitude. We also prove a counting version of a previous theorem on rainbow triangles under a color neighborhood union condition due to Broersma et al., and an asymptotically tight color degree condition forcing a colored friendship subgraph (i.e., rainbow triangles sharing a common vertex).
Funder
National Natural Science Foundation of China
Reference23 articles.
1. Rainbow triangles and the Caccetta‐Häggkvist conjecture
2. Paths and cycles in colored graphs;Broersma H. J.;Australas. J. Combin,2005
3. Rainbow cycles in edge-colored graphs
4. Long rainbow paths and rainbow cycles in edge colored graphs—A survey;Chen H.;Appl. Math. Comput,2018