Total Vertex-Irregularity Labelings for Subdivision of Several Classes of Trees

AbstractMotivated by the notion of the irregularity strength of a graph introduced by Chartrand et al. [3] in 1988 and various kind of other total labelings, Baca et al. [1] introduced the total vertex irregularity strength of a graph.In 2010, Nurdin, Baskoro, Salman and Gaos[5] determined the total vertex irregularity strength for various types of trees, namely complete k–ary trees, a subdivision of stars, and subdivision of particular types of caterpillars. In other paper[6], they conjectured that the total vertex irregularity strength of any tree T is only determined by the number of vertices of degree 1, 2, and 3 in T . In this paper, we attempt to verify this conjecture by considering a subdivision of several types of trees, namely caterpillars, firecrackers, and amalgamation of stars

I-magic labelings of cubic trees and n-caterpillars

Let G be a graph with q edges, then an edge labeling L: E → {1, 2,.., q} is a bijection from the set of edges E to the set of natural numbers less than or equal to q. A graph is said to be I-magic if there exists an edge labeling such that the sum of all edge labels incident to each internal vertex has the same value, and this value is called the magic index t, while the labeling is called an I-magic labeling. It has been conjectured that all cubic trees are I-magic. In this paper, we develop methods for finding I-magic labelings and determine boundary conditions for the magic index of given tRees We also classify an infinite subclass of cubic trees which are I-magic, namely cubic caterpillars. Furthermore, we classify all n-caterpillars as I-magic for n ≥ 3

On the Graceful Game

A graceful labeling of a graph $G$ with $m$ edges consists of labeling the vertices of $G$ with distinct integers from $0$ to $m$ such that, when each edge is assigned as induced label the absolute difference of the labels of its endpoints, all induced edge labels are distinct. Rosa established two well known conjectures: all trees are graceful (1966) and all triangular cacti are graceful (1988). In order to contribute to both conjectures we study graceful labelings in the context of graph games. The Graceful game was introduced by Tuza in 2017 as a two-players game on a connected graph in which the players Alice and Bob take turns labeling the vertices with distinct integers from 0 to $m$. Alice's goal is to gracefully label the graph as Bob's goal is to prevent it from happening. In this work, we study winning strategies for Alice and Bob in complete graphs, paths, cycles, complete bipartite graphs, caterpillars, prisms, wheels, helms, webs, gear graphs, hypercubes and some powers of paths

Vertex-magic Labeling of Trees and Forests

A vertex-magic total labeling of a graph G(V,E) is a one-to-one map λ from E ∪ V onto the integers {1, 2, . . . , |E| + |V|} such that λ(x) + Σ λ(xy) where the sum is over all vertices y adjacent to x, is a constant, independent of the choice of vertex x. In this paper we examine the existence of vertex-magic total labelings of trees and forests. The situation is quite different from the conjectured behavior of edge-magic total labelings of these graphs. We pay special attention to the case of so-called galaxies, forests in which every component tree is a star

Optimal radio labelings of graphs

Let $\mathbb{N}$ be the set of positive integers. A radio labeling of a graph $G$ is a mapping $\varphi : V(G) \rightarrow \mathbb{N} \cup \{0\}$ such that the inequality $|\varphi(u)-\varphi(v)| \geq diam(G) + 1 - d(u,v)$ holds for every pair of distinct vertices $u,v$ of $G$, where $diam(G)$ and $d(u,v)$ are the diameter of $G$ and distance between $u$ and $v$ in $G$, respectively. The radio number $rn(G)$ of $G$ is the smallest number $k$ such that $G$ has radio labeling $\varphi$ with $\max\{\varphi(v) : v \in V(G)\}$ = $k$. Das et al. [Discrete Math. $\mathbf{340}$(2017) 855-861] gave a technique to find a lower bound for the radio number of graphs. In [Algorithms and Discrete Applied Mathematics: CALDAM 2019, Lecture Notes in Computer Science $\mathbf{11394}$, springer, Cham, 2019, 161-173], Bantva modified this technique for finding an improved lower bound on the radio number of graphs and gave a necessary and sufficient condition to achieve the improved lower bound. In this paper, one more useful necessary and sufficient condition to achieve the improved lower bound for the radio number of graphs is given. Using this result, the radio number of the Cartesian product of a path and a wheel graphs is determined.Comment: 12 pages, This is the final version accepted in Discrete Mathematics Letters Journa

Counting real rational functions with all real critical values

We study the number of real rational degree n functions (considered up to linear fractional transformations of the independent variable) with a given set of 2n-2 distinct real critical values. We present a combinatorial reformulation of this number and pose several related questions.Comment: 12 pages (AMSTEX), 3 picture