Aug 2026· Discrete Mathematics, Algorithms and Applications (DMAA)· 0 citations
Abstract
Let [Formula: see text] be a graph with no isolated vertex. A dominated coloring of [Formula: see text] is a proper coloring of [Formula: see text] such that each color class is dominated by at least one vertex. The minimum number of colors needed for a dominated coloring of [Formula: see text] is called the dominated chromatic number of [Formula: see text], denoted by [Formula: see text]. In this paper, we study the dominated chromatic number of central graphs. We obtain some tight bounds for the dominated chromatic number of a central graph [Formula: see text] in terms of some invariants of the graph [Formula: see text]. Also we characterize the dominated chromatic number of the central graph of some families of graphs such as star graphs, path graphs, spider graphs, cycle graphs, wheel graphs, complete graphs, complete bipartite graphs and friendship graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the dominated chromatic number of central graphs.
A 2-distance [Formula: see text]-coloring of a graph is a proper coloring of the vertices of the graph using [Formula: see text] colors such that any two vertices at distance two or less get distinct colors. The 2-distance chromatic number of a graph [Formula: see text], denoted as [Formula: see text], is the minimum i...
Sara Al Hajjar· Discrete Mathematics, Algori...· 0 citations
The eccentricity of any vertex [Formula: see text] in a connected graph [Formula: see text] is the length of the largest distance from [Formula: see text] to any other vertex in [Formula: see text]. The eccentric graph of any graph [Formula: see text], denoted by [Formula: see text], is a graph with the same vertex set...
H. Deepika, T. A. Mangam· Journal of Interconnection N...· 0 citations
Let [Formula: see text] and [Formula: see text] respectively denote the chromatic number and clique number of a graph [Formula: see text]. In this paper, we present our contributions to two open problems on the coloring of [Formula: see text]-free graphs. A question posed by Gyárfás (1987) asks for the smallest [Formul...
C. U. Angeliya, S. Choudum, Mayamma Joseph· Asian-European Journal of Ma...· 0 citations
A hereditary class [Formula: see text] of graphs is [Formula: see text]-bounded if there is a [Formula: see text]-binding function, say [Formula: see text], such that [Formula: see text], for every [Formula: see text], where [Formula: see text] denotes the chromatic (clique) number of [Formula: see text]. A [Formula: s...
Li Zhang, Xia Hong· Discrete Mathematics, Algori...· 0 citations
Graphs whose edge rings have regularity two are not completely characterized. We, therefore, study the minimal graded free resolution of edge rings of certain families of graphs, such as the general barbell graph [Formula: see text] (or [Formula: see text]-barbell graph) and the complete [Formula: see text]-sunlet grap...
Shahnawaz Ahmad Rather, S. Pirzada· Algebra Colloquium· 0 citations
Let [Formula: see text] be a simple graph with [Formula: see text], minimum degree [Formula: see text], and domination number [Formula: see text]. The Minimum Dominating Minimum Degree Matrix, denoted by [Formula: see text], is introduced as a domination–constrained refinement of the classical minimum degree matrix, wh...
