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.

翻译：图G的顶点子集S称为G的控制集，若每个不在S中的顶点均与S中某顶点相邻。G的控制数$\gamma$(G)是G中最小控制集的基数。在2008年一篇突破性论文中，Löwenstein与Rautenbach证明：若G是n阶立方图且围长至少为83，则$\gamma$(G) $\le$ n/3。一个自然的问题是能否降低这一围长条件。该问题催生了两个关于立方图控制的1/3猜想。第一个猜想由Verstraete于2010年提出：若G是n顶点且围长至少为6的立方图，则$\gamma$(G) $\le$ n/3。第二个猜想最初由Kostochka于2009年以问题形式提出：若G是n阶立方二分图，则$\gamma$(G) $\le$ n/3。本文中，当图中不含7-环和8-环时，我们证明了Verstraete猜想；对于不含4-环和8-环的二分图情形，我们证明了Kostochka的相关猜想。