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的顶点集合S若满足G中每个不在S中的顶点均与S中某个顶点相邻，则称S为G的一个支配集。图G的支配数，记作$\gamma$(G)，是指G中支配集的最小基数。在2008年的一项突破性工作中，Löwenstein与Rautenbach证明了：若G是阶数为n且围长至少为83的三次图，则$\gamma$(G) $\le$ n/3。一个自然的问题是此围长条件能否进一步降低。该问题催生出了关于三次图支配数的两个1/3猜想。第一个猜想由Verstraete于2010年提出，其断言：若G是n个顶点且围长至少为6的三次图，则$\gamma$(G) $\le$ n/3。第二个猜想最初由Kostochka于2009年以问题的形式提出，其断言：若G是阶数为n的三次二分图，则$\gamma$(G) $\le$ n/3。本文中，我们证明了当图中不存在7-圈和8-圈时Verstraete猜想成立，并在二分图中不存在4-圈和8-圈时证明了Kostochka的相关猜想。