# More relations between λ-labeling and Hamiltonian paths with emphasis on line graph of bipartite multigraphs

This paper deals with the λ-labeling and L(2,1)-coloring of simple graphs. A λ-labeling of a graph G is any labeling of the vertices of G with different labels such that any two adjacent vertices receive labels which differ at least two. Also an L(2,1)-coloring of G is any labeling of the vertices of G such that any two adjacent vertices receive labels which differ at least two and any two vertices with distance two receive distinct labels. Assume that a partial λ-labeling f is given in a graph G. A general question is whether f can be extended to a λ-labeling of G. We show that the extension is feasible if and only if a Hamiltonian path consistent with some distance constraints exists in the complement of G. Then we consider line graph of bipartite multigraphs and determine the minimum number of labels in L(2,1)-coloring and λ-labeling of these graphs. In fact we obtain easily computable formulas for the path covering number and the maximum path of the complement of these graphs. We obtain a polynomial time algorithm which generates all Hamiltonian paths in the related graphs. A special case is the Cartesian product graph K_n K_n and the generation of λ-squares.

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