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An edge set $S$ of a connected graph $G$ is called an anti-Kekul\'e set if $G-S$ is connected and has no perfect matchings, where $G-S$ denotes the subgraph obtained by deleting all edges in $S$ from $G$. The anti-Kekul\'e number of a graph $G$, denoted by $ak(G)$, is the cardinality of a smallest anti-Kekul\'e set of $G$. It is NP-complete to find the smallest anti-Kekul\'e set of a graph. In this paper, we show that the anti-Kekul\'{e} number of a 2-connected cubic graph is either 3 or 4, and the anti-Kekul\'{e} number of a connected cubic bipartite graph is always equal to 4. Furthermore, a polynomial time algorithm is given to find all smallest anti-Kekul\'{e} sets of a connected cubic graph., Comment: 14 pages, 3 figures

An L(2, 1)-labeling of a graph is an assignment of nonnegative integers to the vertices of G such that adjacent vertices receive numbers differed by at least 2, and vertices at distance 2 are assigned distinct numbers. The L(2, 1)-labeling number is the minimum range of labels over all such labeling. It was shown by Griggs and Yeh [Labelling graphs with a condition at distance 2, SIAM J. Discrete Math. 5(1992), 586-595] that the L(2, 1)-labeling number of a tree is either \D+ 1 or \D + 2. In this paper, we give a complete characterization of L(2, 1)-labeling number for trees with maximum degree 3.