The domination game is played on a graph G by two players, Dominator and Staller, who alternately chooses a vertex of G in such a way that at least one new vertex is dominated. The game ends when all vertices are dominated. Dom- inator aims to finish the game in as few moves as possible while Staller aims to finish the game in as many moves as possible. The game domination number γg (G) (respectively γʹg (G)) is the total number of moves both players use in a game which Dominator (respectively Staller) starts and both players use optimal strategies. In this paper we determine the game domination numbers of a disjoint union of paths and cycles.

The domination game is played on a graph by two players, Dominator and Staller, who alternately choose a vertex of G. Dominator aims to finish the game in as few turns as possible while Staller aims to finish the game in as many turns as possible. The game ends when all vertices are dominated. The game domination number, denoted by ${\mathsf{\gamma }}_g\left(G\right)$ (respectively ${\mathsf{\gamma }}_g^{\prime }\left(G\right)$ ), is the total number of turns when both players play optimally and when Dominator (respectively Staller) starts the game. In this paper, we study a version of this game where the set of chosen vertices is always independent. This version turns out to be another game known as the competition-independence game. The competition-independence game is played on a graph by two players, Diminisher and Sweller. They take turns in constructing maximal independent set M, where Diminisher tries to minimize $\left|M\right|$ and Sweller tries to maximize $\left|M\right|$ . Note that, actually, it is the domination game in which the set of played vertices is independent. The competition-independence number, denoted by $I_d\left(G\right)$ (respectively $I_s\left(G\right)$ ) is the optimal size of the final independent set in the competition-independence game if Diminisher (respectively Sweller) starts the game. In this paper, we check whether some well-known results in the domination game hold for the competition-independence game. We compare the competition-independence numbers to the game domination numbers. Moreover, we provide a family of graphs such that many parameters are equal. Finally, we present a realization result on the competition-independence numbers.

An equitable k-coloring of a graph G is a proper k-coloring of G such that the sizes of any two color classes differ by at most one. An equitable $(q,r)$-tree-coloring of a graph G is an equitable q-coloring of G such that the subgraph induced by each color class is a forest of maximum degree at most r. Let the strong equitable vertex r-arboricity of a graph $G,$ denoted by $v{a}_r^{\equiv }\left(G\right)$, be the minimum p such that G has an equitable $(q,r)$-tree-coloring for every $q\ge p.$ The values of $v{a}_1^{\equiv }\left(K_{n,n}\right)$ were investigated by Tao and Lin and Wu, Zhang, and Li where exact values of $v{a}_1^{\equiv }\left(K_{n,n}\right)$ were found in some special cases. In this paper, we extend their results by giving the exact values of $v{a}_1^{\equiv }\left(K_{n,n}\right)$ for all cases. In the process, we introduce a new function related to an equitable coloring and obtain a more general result by determining the exact value of each $v{a}_1^{\equiv }\left(K_{m,n}\right)$ and $v{a}_1^{\equiv }\left(G\right)$ where G is a balanced complete k-partite graph $K_{n,\dots ,n}$. Both complete bipartite graphs $K_{m,n}$ and balanced complete k-partite graphs $K_{n,\dots ,n}$ are symmetry in several aspects and also studied broadly. For the other aspect of symmetry, by the definition of equitable k-coloring of graphs, in a specific case that the number of colors divides the number of vertices of graph, we can say that the graph is a balanced k-partite graph.