E. Sampathkumar

Strong weak domination and domination balance in a graph

Abstract Let G = (V, E) be a graph and u, v ∈ V. Then, ustrongly dominates v and v weakly dominates u if (i) uv ∈ E and (ii) deg u ⩾ deg v. A set D ⊂ V is a strong-dominating set (sd-set) of G if every vertex in V − D is strongly dominated by at least one vertex in D. Similarly, a weak-dominating set (wd-set) is defined. The strong (weak) domination number γs (γw) of G is the minimum cardinality of an sd-set (wd-set). Besides investigating some relationship of γs and γw with other known parameters of G, some bounds are obtained. A graph G is domination balanced if there exists an sd-set D1 and a wd-set D2 such that D1 ∩ D2 = 0. A study of domination balanced graphs is initiated.

3-consecutive c-colorings of graphs

A 3-consecutive C-coloring of a graph G = (V;E) is a mapping ’ : V !N such that every path on three vertices has at most two colors. We prove general estimates on the maximum number „ ´3CC(G) of colors in a 3-consecutive C-coloring of G, and characterize the structure of connected graphs with „ ´3CC(G) ‚ k for k = 3 and k = 4.

Vertex coloring without large polychromatic stars

Abstract Given an integer k ≥ 2 , we consider vertex colorings of graphs in which no k -star subgraph S k = K 1 , k is polychromatic. Equivalently, in a star- [ k ] -coloring the closed neighborhood N [ v ] of each vertex v can have at most k different colors on its vertices. The maximum number of colors that can be used in a star- [ k ] -coloring of graph G is denoted by χ k ⋆ ( G ) and is termed the star- [ k ] upper chromatic number of G . We establish some lower and upper bounds on χ k ⋆ ( G ) , and prove an analogue of the Nordhaus–Gaddum theorem. Moreover, a constant upper bound (depending only on k ) can be given for χ k ⋆ ( G ) , provided that the complement G ¯ admits a star- [ k ] -coloring with more than k colors.

Improper C-colorings of graphs

For an integer k ? 1 , the k -improper upper chromatic number ? ? k - imp ( G ) of a graph G is introduced here as the maximum number of colors permitted to color the vertices of G such that, for any vertex v in G , at most k vertices in the neighborhood N ( v ) of v receive colors different from that of v . The exact value of ? ? k - imp is determined for several types of graphs, and general estimates are given in terms of various graph invariants, e.g.?minimum and maximum degree, vertex covering number, domination number and neighborhood number. Along with bounds on ? ? k - imp for Cartesian products of graphs, exact results are found for hypercubes. Also, the analogue of the Nordhaus-Gaddum theorem is proved. Moreover, the algorithmic complexity of determining ? ? k - imp is studied, and structural correspondence between k -improper C-colorings and certain kinds of edge cuts is shown.

Generalizations of independence and chromatic numbers of a graph

Abstract Let G=(V,E) be a graph and k⩾2 be an integer. A set S⊂V is k-independent if every component in the subgraph

The least point covering and domination numbers of a graph

Abstract A set S ⊂ V of a graph G = ( V , E ) is a total point cover (t.p.c.) if S is a point cover containing all isolates of G , if any. The number α t ( G ) is the minimum cardinality of a t.p.c. A t.p.c. S is a least point cover (l.p.c.) if α t (〈 S 〉)⩽ α t (〈 S 1 〉) for any t.p.c. S 1 , where 〈 S 〉 is the subgraph induced by S . The least point covering number α 1 ( G ) of G is the minimum cardinality of a l.p.c. A dominating set D of G is a least dominating set (l.d.s.) if γ (〈 D 〉) ⩽ γ (〈 D 1 〉) for any dominating set D 1 (γ denotes domination number). The least domination number γ 1 ( G ) of G is the minimum cardinality of a l.d.s. If γ t is the total domination number, we prove among 0other things: (i) γ 1 ⩽ γ t , and (ii) for a tree, γ 1 ⩽ α 1 . Conjectures . For any graph G of order p ⩾ 2, (1) γ 1 ⩽ α 1 , (2) γ 1 ⩽ 3p 5 , if G is connected.

A generalization of chromatic index

Abstract Let G = ( V , E ) be a graph and k ⩾ 2 an integer. The general chromatic index χ′ k ( G ) of G is the minimum order of a partition P of E such that for any set F in P every component in the subgraph 〈 F 〉 induced sby F has size at most k - 1. This paper initiates a study of χ′ k ( G ) and generalizes some known results on chromatic index.