  A set S of vertices in a graph G is a dominating set of G if every vertex not
in S is adjacent to a vertex in S . The domination number of G, denoted by
$\gamma$(G), is the minimum cardinality of a dominating set in G. In a
breakthrough paper in 2008, L{\"o}wenstein and Rautenbach proved that if G is a
cubic graph of order n and girth at least 83, then $\gamma$(G) $\le$ n/3. A
natural question is if this girth condition can be lowered. The question gave
birth to two 1/3-conjectures for domination in cubic graphs. The first
conjecture, posed by Verstraete in 2010, states that if G is a cubic graph on n
vertices with girth at least 6, then $\gamma$(G) $\le$ n/3. The second
conjecture, first posed as a question by Kostochka in 2009, states that if G is
a cubic, bipartite graph of order n, then $\gamma$(G) $\le$n/3. In this paper,
we prove Verstraete's conjecture when there is no 7-cycle and no 8-cycle, and
we prove the Kostochka's related conjecture for bipartite graphs when there is
no 4-cycle and no 8-cycle.
