A graph tessellation is a partition of the vertices of the graph into cliques and a graph tessellation cover is a set of graph tessellations that covers the edges of the graph. A graph is 3-tessellable if it has a tessellation cover with three tessellations. The study of graph tessellations is important in quantum computation because the evolution operator of the staggered quantum walk model is obtained from a graph tessellation cover. In this work we establish a characterization on the smallest tessellation cover of a graph G using the chromatic number of its clique graph χ(K(G)) by showing that a diamond-free graph G is 3tessellable if and only if χ(K(G)) ≤ 4 and there is a vertex coloring assignment of K(G) with a special property. As a consequence of such characterization, we obtain a hardness proof for determining if a line graph of a triangle-free graph is 3-tessellable. Moreover, we introduce a special type of edge coloring of a triangle-free graph G which corresponds to a tessellation cover of its line graph. This hardness proof allows us to establish the N P -completeness of this new coloring problem for triangle-free graphs.

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The staggered model is a discrete-time quantum walk model, whose evolution operator is obtained from a set of graph tessellations. We use this model to analyze the standard deviation of the probability distribution on a line of diamonds. The results help to understand the dynamics of quantum walks and it can be used as tools for building quantum algorithms.