Abstract. In this paper, we are mainly concerned with the enumeration of (2k + 1, 2k + 3)-core partitions with distinct parts. We derive the number and the largest size of such partitions, confirming two conjectures posed by Straub.

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Given two graphs G

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.

In 2011, Caro et al. introduced the monochromatic connection of graphs. An edge-coloring of a connected graph G is called a monochromatically connecting (MC-coloring, for short) if there is a monochromatic path joining any two vertices. The monochromatic connection number mc(G) of a graph G is the maximum integer k such that there is a k-edge-coloring, which is an MC-coloring of G. Clearly, a monochromatic spanning tree can monochromatically connect any two vertices. So for a graph G of order n and size m, mc(G) ≥ m − n + 2. Caro et al. proved that both triangle-free graphs and graphs of diameter at least three meet the lower bound.In this paper, we consider the monochromatic connectivity of graphs containing triangles which meet the lower bound too. Also, in order to study the graphs of diameter two, we present the formula for the monochromatic connectivity of join graphs. This will be helpful to solve the problem for graphs of diameter two.