Harmonious Coloring on Subclasses of Colinear Graphs
نویسندگان
چکیده
Given a simple graph G, a harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number is the least integer k for which G admits a harmonious coloring with k colors. Extending previous NP-completeness results of the harmonious coloring problem on subclasses of chordal and co-chordal graphs, we prove that the problem remains NP-complete for split undirected path graphs; we also prove that the problem is NP-complete for colinear graphs by showing that split undirected path graphs form a subclass of colinear graphs. Moreover, we provide a polynomial solution for the harmonious coloring problem for the class of split strongly chordal graphs, the interest of which lies on the fact that the problem has been proved to be NP-complete on both split and strongly chordal graphs.
منابع مشابه
NP-completeness results for some problems on subclasses of bipartite and chordal graphs
Extending previous NP-completeness results for the harmonious coloring problem and the pair-complete coloring problem on trees, bipartite graphs and cographs, we prove that these problems are also NP-complete on connected bipartite permutation graphs. We also study the k-path partition problem and, motivated by a recent work of Steiner [G. Steiner, On the k-path partition of graphs, Theoret. Co...
متن کاملColinear Coloring and Colinear Graphs∗
Motivated by the notion of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology, and the framework through which it was studied, we introduce the colinear coloring on graphs. We provide an upper bound for the chromatic number χ(G), for any graph G, and show that G can be colinearly colored in polynomial time by proposing a simple algorithm. The colin...
متن کاملThe harmonious coloring problem is NP-complete for interval and permutation graphs
In this paper, we prove that the harmonious coloring problem is NP-complete for connected interval and permutation graphs. Given a simple graph G, a harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number is the least integer k for which G admits a harmonious coloring with k colors. Extending previo...
متن کاملColinear Coloring on Graphs
Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology, and the framework through which it was studied, we introduce the colinear coloring on graphs. We provide an upper bound for the chromatic number χ(G), for any graph G, and show that G can be colinearly colored in polynomial time by proposing a simple algorithm. The c...
متن کاملHarmonious and achromatic colorings of fragmentable hypergraphs
A harmonious coloring of a k-uniform hypergraphH is a rainbow vertex coloring such that each k-set of colors appears on at most one edge. A rainbow coloring of H is achromatic if each k-set of colors appears on at least one edge. The harmonious (resp. achromatic) number of H , denoted by h(H) (resp. ψ(H)) is the minimum (resp. maximum) possible number of colors in a harmonious (resp. achromatic...
متن کامل