Narrow Results

A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by $\phi \left(G\right)$, is the maximal integer k such that G has a b-coloring with k colors. In this paper, the b-chromatic numbers of the coronas of cycles, star graphs and wheel graphs with different numbers of vertices, respectively, are obtained. Also the bounds for the b-chromatic number of corona of any two graphs is discussed.

The aim of this study is to compute the edge metric dimension of some subdivision of the wheel graphs. In particular, we determine and compare the metric and edge metric dimensions of the graphs obtained after the cycle, spoke and barycentric subdivisions of the wheel graph. Furthermore, some families of graphs have been constructed through subdivision process for which $\mathrm{\text{dim}}({{𝒲}{𝒞}{𝒫}}_{n,1})<{\mathrm{\text{dim}}}_e({{𝒲}{𝒞}{𝒫}}_{n,1})$, and also $\mathrm{\text{dim}}({{𝒲}{𝒞}{𝒴}}_{n,1})={\mathrm{\text{dim}}}_e({{𝒲}{𝒞}{𝒴}}_{n,1})$ which partially answer a question in [A. Kelenc, N. Tratnik and I. G. Yero, Uniquely identifying the edges of a graph: The edge metric dimension, Discrete Appl. Math.251 (2018) 204–220].

A graph $G$ with $p$ vertices and $q$ edges has antimagic labeling if there is a bijection from the edge set of the graph to the label set $\{1,2,\dots ,q\}$ such that $p$ vertices must have distinct vertex sums, where the vertex sums are determined by adding up all the edge labels incident to each vertex $v$ in $V(G)$. Hartsfield and Ringel conjectured that every connected graph is antimagic, except $P_2$. In this study, we identified a class of connected graphs that lend credence to this conjecture. In this paper, we proved that the tensor product of a Wheel Graph, Helm Graph, and Flower graph with Star is antimagic.

The strong edge coloring of a graph G is a proper edge coloring that assigns a different color to any two edges which are at most two edges apart. The minimum number of color classes that contribute to such a proper coloring is said to be the strong chromatic index of G. This paper defines the strong chromatic index for the generalized Jahangir graphs and the generalized Helm graphs.