Totally irregular total labeling of some caterpillar graphs

Let G be a simple, connected and undirected graph with vertex set V and edge set E. A total k-labeling is defined as totally irregular total k-labeling if the weights of any two different both vertices and edges are distinct. The weight of vertex x is defined as , while the weight of edge xy is . A minimum k for which G has totally irregular total k-labeling is mentioned as total irregularity strength of G and denoted by ts(G). This paper contains investigation of totally irregular total k-labeling and determination of their total irregularity strengths for caterpillar graphs with each internal vertex between two stars has degree three. The results are and for n > 4:

This paper deals with the labeling of vertices and edges of a graph. Let G be a graph with vertex set V and edge set E, where |V| be the number of vertices and |E| edges of G. The two bijection methods for which we have designed algorithms are as follows. Initially A bijection λ 1:V∪E → {1, 2…|V| + |E|} is called a Vertex-Magic Total Labeling (VMTL) if there is a vertex magic constant vk such that the weight of vertex m is, λ 1(m) + Σ n εA (m) λ 1(mn) = vk, ∀ m εv Where A(m) is the set of vertices adjacent to x. In the similar fashion the bijection λ 2:V∪E → {1, 2…|V| + |E|} is called Edge-Magic Total Labeling (EMTL) if there is a edge magic constant ek such that the weight of an edge es(mn), λ 2(m) + λ 2(n) + λ 2 e(mn) = ek, ∀ e εE. A resultant Graph which consists of both VMTL and EMTL are said to be Total Magic Labeling (TML) for different vertex magic constant and edge magic constant values. Baker and Sawada proposed algorithms to find VMTLs on cycles and wheels. In this paper we enhanced these algorithms and also we proposed new algorithms to generate EMTLs and TMLs of cycles and wheels. We used the concept variations and sum set sequences to produce VMTLs and EMTLs on cycles and wheels. Also we designed modules to identify TML’s.KeywordsMagic labelingMagic labeling algorithmsMagic constantCyclesWheelsVertex Magic Total LabelingEdge Magic Total LabelingTotal Magic Labeling etc

Total labellings, irregularity strengths and colourings

We introduce the notion of total distance irregular labeling, called the local total distance irregular labeling. All edges and vertices are labeled with positive integers 1 to k such that the weight calculated at the vertices induces a vertex coloring if two adjacent vertices has different weight. The weight of a vertex $u\in V(G)$ is defined as the sum of the labels of all vertices adjacent and edges incident to $u$ (distance $1$ from $u$). The minimum cardinality of the largest label over all such irregular assignment is called the local total distance irregularity strength, denoted by $tdis_l(G)$. In this paper, we established the lower bound of the local total distance irregularity strength of graphs $G$ and determined exact values of some classes of graphs namely path, cycle, star, bipartite complete, fan and sun graph.

Let G be a graph which has V(G) as a vertex set and E(G) as an edge set. The G graph contain subgraph H that isomorphic with Hj, j = 1, 2, … s, if every e ∈ E(G) include only in one of the edge set of H subgraph (e ∈ E(H)). The total l-labeling is a graph labeling that give label positive integer number until l into vertices and edges. The total H-irregular labeling is a total l-labeling with condition that the sum of vertex labels and edge labels in two distinct subgraphs H 1 and H 2 isomorphic to H is different. We define H-weight as wtφ (H) = ∑ v∈V(H) φ(v) + ∑ e∈E(H) φ(e), for the subgraph H ⊆ G under the total l-labeling (φ). The total H-irregularity strength of G graph (ths(G, H)) is the smallest l value in label of G graph has total H-irregular labeling. We used plane graphs such as triangular ladder graph and windmill graph.

Let G(V, E) be a finite, simple and undirected graph with a vertex set V and an edge set E. An edge irregular total k-labeling is a map f : V ⋃ E → {1, 2, …, k} such that for any two different edges xy and x′y′ in E, ω(xy) ≠ ω(x′y′) where ω(xy) = f(x) + f(y) + f(xy). The minimum k for which the graph G admits an edge irregular total k-labeling is called the total edge irregularity strength of G, denoted by tes(G). We have constructed the formula of an edge irregular total k-labelling and determined the total edge irregularity strength of book graphs and double book graphs. In this paper, we construct an edge irregular total k-labeling that can be used for book graphs, double book graphs, and triple book graphs. We also show the exact value of the total edge irregularity strength of triple book graphs.

A (2, 1)-total labeling of a graph G is an assignment f from the vertex set V(G) and the edge set E(G) to the set {0, 1, ..., k} of nonnegative integers such that |f(x) - f(y)| ≥ 2 if x is a vertex and y is an edge incident to x, and |f(x) - f(y)| ≥ 1 if x and y are a pair of adjacent vertices or a pair of adjacent edges, for all x and y in V(G) ∪ E(G). The (2, 1)-total labeling number λ2T(G) of G is defined as the minimum k among all possible assignments. In [D. Chen and W. Wang. (2,1)-Total labelling of outerplanar graphs. Discr. Appl. Math. 155 (2007)], it was conjectured that all outerplanar graphs G satisfy λ2T(G) ≤ Δ(G)+2, where Δ(G) is the maximum degree of G, while they also showed that it is true for G with Δ(G) ≥ 5. In this paper, we solve their conjecture completely, by proving that λ2T(G) ≤ Δ(G)+2 even in the case of Δ(G) ≤ 4.

Total edge irregularity strength of complete graphs and complete bipartite graphs

A total labeling Φ : V (G) ∪ E(G) → {1, 2, ... , k} is called a vertex irregular total k-labeling of a graph G if different vertices in G have different weights. The weight of a vertex is defined as the sum of the labels of its incident edges and the label of that vertex. The minimum k for which the graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, denoted by tvs(G). In this paper we deal with the total vertex irregularity strength of uniform theta graphs and centralized uniform theta graphs. Theta graph is a closer representation of bipolar electric or magnetic fields so labeling of various theta graphs can help the law of physics in future.

An edge irregular total k-labeling f : V ∪ E → {1, 2, 3,...,k} of a graph G = (V, E) is a labeling of vertices and edges of G in such a way thatfor any two different edges uv and u'v' their weights f (u) + f (uv) + f (v) and f (u') + f (u'v') + f (v') are distinct. The total edge irregularity strength tes(G) is defined as the minimum k for which the graph G has an edge irregular total k-labeling. In this paper, we determine the total edge irregularity strength of disjoint union of p isomorphic double wheel graphs and disjoint union of p consecutive non-isomorphic double wheel graphs.

All graph in this paper is simple and connected graph where $V(G)$ is vertex set and $E(G)$ is edge set. Let function $f : V(G)\longrightarrow \{0, 2,..., 2k_v\}$ as vertex labeling and a function $f: E(G)\longrightarrow \{1, 2,..., k_e\}$ as edge labeling where $k=max\{2k_v,k_e\}$ for $k_v,k_e$ are natural number. The weight of edge $ u,v\in E(G) $ under $f$ is $w(u)=f(u)+ \Sigma_{uv \in V(G)} f(uv)$. In other words, the function $f$ is called local edge irregular reflexive labeling if every two adjacent edges has distinct weight and weight of a edge is defined as the sum of the labels of edge and the labels of all vertex incident this edge When we assign each edge of $G$ with a color of the edge weight $w(uv)$, thus we say the graph $G$ admits a local edge irregular reflexive coloring. The minimum number of colors produced from local edge irregular reflexive coloring of graph $G$ is reflexive local irregular chromatic number denoted by $\chi_{lrecs}(G).$ Furthermore, the minimum $k$ required such that $\chi_{lrecs}(G)=\chi(G)$ is called a local reflexive edge color strength, denoted by \emph{lrecs}$(G)$. In this paper, we learn about the local edge irregular reflexive coloring and obtain \emph{lrecs}$(G)$ of planar related graphs.

In graph theory many useful mathematical disciplines, but the extremely useful one is graph labeling. A graph labeling is used in many applications in various fields like X-ray crystallography, computer science, data base management, coding theory, astronomy and physics. A labeling of edges and vertices of a simple graph G(V, E) by a mapping ¥ : V(G) ∪ E(G) → {1, 2, 3, … , ℜ} provided that any two different edges have distinct weights is called an edge irregular total ℜ -labeling. If ℜ is minimum and G admits an edge irregular total ℜ-labeling, then it is called the total edge irregularity strength (TEIS) denoted by tes(G). In this paper, we have introduced three definitions: the quintet snake graph QS n , the double quintet snake graph D(QS n) and the m–multiple quintet snake graph M(QS n). Moreover, the exact value of TEIS for the new graphs are investigated.

Let <em>G </em>= (<em>V</em>(<em>G</em>),<em>E</em>(<em>G</em>)) be a graph and <em>k</em> be a positive integer. A total <em>k</em>-labeling of <em>G</em> is a map <em>f</em> : <em>V</em> ∪ <em>E</em> → {1,2,3,...,<em>k</em>}. The vertex weight <em>v</em> under the labeling <em>f</em> is denoted by w_<em>f</em>(<em>v</em>) and defined by <em>w</em>_<em>f</em>(<em>v</em>) = <em>f</em>(<em>v</em>) + \sum_{uv \in{E(G)}} {<em>f</em>(<em>uv</em>)}. A total <em>k</em>-labeling of <em>G</em> is called vertex irregular if there are no two vertices with the same weight. The total vertex irregularity strength of <em>G</em>, denoted by <em>tvs</em>(<em>G</em>), is the minimum <em>k</em> such that <em>G</em> has a vertex irregular total <em>k</em>-labeling. This labelings were introduced by Baca, Jendrol, Miller, and Ryan in 2007. Let <em>G</em> and <em>H</em> be two connected graphs. Let <em>o</em> be a vertex of <em>H</em>. The comb product between <em>G</em> and <em>H</em>, denoted by <em>G </em>\rhd_o <em>H</em>, is a graph obtained by taking one copy of <em>G</em> and |<em>V</em>(<em>G</em>)| copies of <em>H</em> and grafting the i-th copy of <em>H</em> at the vertex <em>o</em> to the i-th vertex of <em>G</em>. In this paper, we determine the total vertex irregularity strength of comb product of two cycles and two stars.

A (p,1)- total labeling of G is an assignment of integers to V(G)∪E(G) such that any two adjacent vertices of G receive distinct integers, any two adjacent edges of G receive distinct integers, and a vertex and its incident edge receive integers that differ by at least p in absolute value. The span of a (p,1)-total labeling is the maximum difference between two labels. The minimum span of a (p,1)-total labeling of G is called the (p,1)-total number and denoted by λ^T_P(G). Let n and k be two positive integers. The graph with vertex sets {u(1),...,u(n)} and {v(1),...,v(n)} and edge sets {u(i)u(i+1)|i=1,2,...,n},{u(i)v(i)|i=1,2,...,n} and {v(i)v(i+k)|i=1,2,...,n;k<n}, where addition is modulo n is called generalized Petersen graph and denoted by P(n,k). In this thesis, we mainly focus on the (2,1)-total labeling of the generalized Petersen graph, and we show that for each pair of positive integer n and k, 1<=n, if n≡0(mod 5) and k≠0(mod 5), then λ^T_2(P(n,k))=5. Moreover, we also prove that λ^T_2(P(n,5))=5 if n≡0(mod 25).

LetG= (V, E)be a finite, simple and undirected graph with a vertex setVand an edge setE. An edge irregular totalk-labelling is a functionf : V ᴗE → {1,2,…,k}such that for any two different edgesxyandx’y’inE, their weights are distinct. The weight of edgexyis the sum of label of edgexy, labels of vertexxand of vertexy. The minimumkfor which the graphGadmits an edge irregular totalk-labelling is called the total edge irregularity strength ofG, denoted bytes(G). We have determined the total edge irregularity strength of book graphs, double book graphs and triple book graphs. In this paper, we show the exact value of the total edge irregularity strength of quadruplet book graphs and quintuplet book graphs.