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For a positive integer k, a radio k-coloring of a simple connected graph G = (V(G), E(G)) is a mapping f : V (G) → { 0 , 1 , 2 , ... } such that | f (u) - f (v) | ≥ k + 1 - d (u , v) for each pair of distinct vertices u and v of G, where d(u, v) is the distance between u and v in G. The span of a radio k-coloring f, rck(f), is the maximum integer assigned by it to some vertex of G. The radio k-chromatic number, rck(G) of G is min{rck(f)}, where the minimum is taken over all radio k-colorings f of G. If k is the diameter of G, then rck(G) is known as the radio number of G. In this paper, we propose an improved upper bound of radio k-chromatic number for a given graph against the other which is due to Saha and Panigrahi (in: Arumugan, Smyth (eds) Combinatorial algorithms (IWOCA 2012). Lecure notes in computer science, vol 7643, Springer, Berlin, 2012). The computational study shows that the proposed algorithm overcomes the previous algorithm. We introduce a polynomial algorithm [differs from the other that is due to Liu and Zhu (SIAM J Discrete Math 19(3):610–621, 2005)] which determines the radio number of the path graph Pn. Finally, we propose a new integer linear programming model for the radio k-coloring problem. The computational study between the proposed algorithm and LINGO solver shows that the proposed algorithm overcomes LINGO solver. [ABSTRACT FROM AUTHOR]

In this paper, we prove that, given a clique-width k-expression of an n-vertex graph, Hamiltonian Cycle can be solved in time n O (k) . This improves the naive algorithm that runs in time n O (k 2) by Espelage et al. (Graph-theoretic concepts in computer science, vol 2204. Springer, Berlin, 2001), and it also matches with the lower bound result by Fomin et al. that, unless the Exponential Time Hypothesis fails, there is no algorithm running in time n o (k) (Fomin et al. in SIAM J Comput 43:1541–1563, 2014). We present a technique of representative sets using two-edge colored multigraphs on k vertices. The essential idea is that, for a two-edge colored multigraph, the existence of an Eulerian trail that uses edges with different colors alternately can be determined by two information: the number of colored edges incident with each vertex, and the connectedness of the multigraph. With this idea, we avoid the bottleneck of the naive algorithm, which stores all the possible multigraphs on k vertices with at most n edges. [ABSTRACT FROM AUTHOR]