Academic literature on the topic 'Triangular Snake Graph'

Let λ:V(Ω)→{1, 2,..., z} be a mapping of vertices of a graph Ω. Let S(x) denote the sum of labels of the neighbors of the vertex x in Ω. If vertex x has degree zero, we put S(x)=0. A mapping λ is categorized as lucky labeling if S(x)=S(y) for every pair of adjacent vertices x and y. The lucky number of graph Ω, denoted by η (Ω), is the least positive integer z used to label vertices to form lucky labeling. In this paper, we demonstrate that different families of comb graph and snake graph are lucky labeled graphs. We also calculate the exact value of the lucky number for the aforementioned graphs.

<p>Let G be a (p, q) graph. Let f : V (G) → {1, 2, . . . , k} be a map where k is an integer 2 ≤ k ≤ p. For each edge uv, assign the label |f(u) − f(v)|. f is called k-difference cordial labeling of G if |vf (i) − vf (j)| ≤ 1 and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labelled with x, ef (1) and ef (0) respectively denote the number of edges labelled with 1 and not labelled with 1. A graph with a k-difference cordial labeling is called a k-difference cordial graph. In this paper we investigate 3-difference cordial labeling behavior of triangular snake, alternate triangular snake, alternate quadrilateral snake, irregular triangular snake, irregular quadrilateral snake, double triangular snake, double quadrilateral snake, double alternate triangular snake, and double alternate quadrilateral snake.</p>

An equitable total-coloring of a graph G is a proper total-coloring such that the number of vertices and edges in any two color classes differ by at most one. In this paper, we determined the equitable total chromatic number for line graph of ladder, slanting ladder, triangular snake, alternate triangular snake, quadrilateral snake and alternate quadrilateral snake Introduction: Graph coloring is a fundamental problem in graph theory with applications in scheduling, networking, and resource allocation. A total-coloring of a graph G is an assignment of colors to both vertices and edges such that adjacent vertices, adjacent edges, and incident vertex-edge pairs receive distinct colors. An equitable total-coloring is a special type of total-coloring where the sizes of any two color classes differ by at most one. The equitable total chromatic number, denoted as is the minimum number of colors required for such a coloring. Objectives: To establish the equitable total chromatic number for the line graph of specific families of structured graphs. To develop systematic coloring techniques for achieving an equitable total- coloring of these graphs. To contribute to the broader study of equitable colorings in graph theory and expand the known results in this domain. Methods: To determine the equitable total chromatic number for the line graphs of the given graph families, we employ the following methodology: Graph Construction: We formally define the structure of the ladder, slanting ladder, triangular snake, alternate triangular snake, quadrilateral snake, and alternate quadrilateral snake, along with their corresponding line graphs. Coloring Strategy: We apply systematic coloring techniques ensuring that adjacent vertices, adjacent edges, and incident vertex-edge pairs receive different colors while maintaining equitable distribution of color classes. Mathematical Analysis: We derive lower bounds for and establish its exact value using combinatorial and structural properties of the graphs. Verification and Proof: We validate the obtained chromatic numbers through case-based analysis and, where applicable, provide rigorous proofs for correctness. Results: The study successfully determines the exact value of the equitable total chromatic number for the line graphs of the considered structured graphs. The results provide new insights into the equitable total-coloring of line graphs of ladder-based and snake-like structures, which are commonly encountered in chemical graph theory and network design problems. Conclusions: This paper establishes the equitable total chromatic number for the line graphs of several structured graphs, contributing to the ongoing research in equitable colorings. The findings demonstrate that the structural properties of the base graphs significantly influence their equitable total chromatic numbers. These results can be extended to other classes of graphs, and future research may explore algorithmic approaches for efficient equitable total-coloring in larger and more complex graph families

A radio labeling of a simple connected graph G = V , E is a function h : V ⟶ N such that h x − h y ≥ diam G + 1 − d x , y , where diam G is the diameter of graph and d(x, y) is the distance between the two vertices. The radio number of G , denoted by rn G , is the minimum span of a radio labeling for G . In this study, the upper bounds for radio number of the triangular snake and the double triangular snake graphs are introduced. The computational results indicate that the presented upper bounds are better than the results of the mathematical model provided by Badr and Moussa in 2020. On the contrary, these proposed upper bounds are better than the results of algorithms presented by Saha and Panigrahi in 2012 and 2018.