The forcing number of graphs with given girth

In this paper, we study a dynamic coloring of the vertices of a graph G that starts with an initial subset S of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set S is called a forcing set of G if, by iteratively applying the forcing process, every vertex in G becomes colored. The forcing number, originally known as the zero forcing number, and denoted F (G), of G is the cardinality of a smallest forcing set of G. We study lower bounds on the forcing number in terms of its minimum degree and girth, where the girth g of a graph is the length of a shortest cycle in the graph. Let G be a graph with minimum degree 2 and girth g 3. Davila and Kenter [Theory and Applications of Graphs, Volume 2, Issue 2, Article 1, 2015] conjecture that F (G) + ( - 2)(g - 3). This conjecture has recently been proven for g 6. The conjecture is also proven when the girth g 7 and the minimum degree is sufficiently large. In particular, it holds when g = 7 and 481, when g = 8 and 649, when g = 9 and 30, and when g = 10 and 34. In this paper, we prove the conjecture for g {7, 8, 9, 10} and for all values of >= 2.