Anti-Ramsey Numbers for Graphs with Independent Cycles

Abstract

An edge-colored graph is called rainbow if all the colors on its edges are distinct. Let ${\cal G}$ be a family of graphs. The anti-Ramsey number $AR(n,{\cal G})$ for ${\cal G}$, introduced by Erdős et al., is the maximum number of colors in an edge coloring of $K_n$ that has no rainbow copy of any graph in ${\cal G}$. In this paper, we determine the anti-Ramsey number $AR(n,\Omega_2)$, where $\Omega_2$ denotes the family of graphs that contain two independent cycles.

Rainbow generalizations of Ramsey theory - a dynamic survey

“…Let Ω k be the set of graphs containing k vertex disjoint disjoint cycles. In [115], the following result was proven along with some general bounds for ar(K n , Ω k ).…”

Anti-Ramsey problems in the generalized Petersen graphs for cycles

“…They presented a close relationship between the anti-Ramsey number and Turán number. Since then, plentiful results were researched for a variety of graphs H, including cycles [1,2,12,13,18,19], cliques [4,17], trees [9,11], and matchings [7,16]. Some other graphs were also considered as the host graphs in anti-Ramsey problems, such as hypergraphs [6], hypecubes [3], complete split graphs [5,14], and triangulations [8,10,15].…”

Anti-Ramsey Numbers in Complete k-Partite Graphs

“…e conjecture is proved completely for all k ≥ 3 in [3] by Montellano-Ballesteros and Neumann-Lara. e anti-Ramsey numbers for some other special graph classes in complete graphs have also been studied, including independent cycles [4], stars [5], spanning trees [6], and matchings [7,8]. e anti-Ramsey problems for rainbow matchings, cycles, and trees in complete bipartite graphs have been studied in [9][10][11].…”