论文标题
对角的广义锤型的度量尺寸
Metric Dimension of a Diagonal Family of Generalized Hamming Graphs
论文作者
论文摘要
经典的锤子图是完整图的笛卡尔产品，如果两个坐标在一个坐标上有所不同，则两个顶点相邻。通过与统一Cayley图的连接的动机，我们考虑了一个概括，如果两个顶点没有共同的坐标，则两个顶点相邻。经典锤图的度量尺寸是渐近的，但是即使在超振管的情况下，也很少发现精确的值。相比之下，我们确定了整个对角线家族的度量尺寸为$ 3 $尺寸的概括性锤图。我们的方法是建设性的，并通过首先根据辅助边缘色超图的禁止子图表来表征解析集。
Classical Hamming graphs are Cartesian products of complete graphs, and two vertices are adjacent if they differ in exactly one coordinate. Motivated by connections to unitary Cayley graphs, we consider a generalization where two vertices are adjacent if they have no coordinate in common. Metric dimension of classical Hamming graphs is known asymptotically, but, even in the case of hypercubes, few exact values have been found. In contrast, we determine the metric dimension for the entire diagonal family of $3$-dimensional generalized Hamming graphs. Our approach is constructive and made possible by first characterizing resolving sets in terms of forbidden subgraphs of an auxiliary edge-colored hypergraph.
