A conjecture on strong magic labelings of 2-regular graphs

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

In this paper it is shown that if a graph G possesses a spanning subgraph H with a strong vertex magic total labeling (VMTL) and $G-E(H)$ is even‐regular, then G also has a strong VMTL. Among other things, this is used to conclude that all Hamiltonian regular graphs of odd order possess strong VMTLs. A relationship is then demonstrated between regular graphs of even degree and sparse magic squares. We next consider cubic graphs of order $2n$ consisting of two 2‐factors of order n, connected by a 1‐factor (quasi‐prisms). Based on McQuillan’s construction of VMTLs of such 3‐regular graphs, VMTLs are derived for similar regular graphs of any odd degree. Finally, a construction is given for VMTLs of quartic graphs of order $4n+2$ consisting of two cycles of odd order n connected by a 2‐factor (simple quasi‐anti‐prisms), and based on this construction VMTLs are derived for similar regular graphs of any even degree.

Consecutive magic graphs

Two new methods to obtain super vertex-magic total labelings of graphs

A one-to-one map X from E ∪ V onto the integers 1, 2, …, e + v is a vertex-magic total labeling if there is a constant h so that for every vertex x, λ(x) + ∑λ(xy) = h(3.1) where the sum is over all vertices y adjacent to x. So the magic requirement is wt(x) — h for all x. The constant h is called the magic constant for λ. Again, a graph with a vertex-magic total labeling will be called vertex-magic

Objectives: In this paper, we prove that the line digraphs of Cayley digraphs admit super vertex (a,d) antimagic labeling, product anti magic and vertex magic total labeling. Also we have shown the labeling formula to get the same. Findings: If a Cayley digraph admits the above three labelings, then the line digraph of Cayley digraph also admits the same.

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.

This study investigates the existence of discrete and strong discrete labeling for several classes of path and star-related graphs, contributing to the broader field of graph labeling theory. A graph is said to admit a discrete labeling if its vertices can be assigned binary labels (0 and 1), such that each edge receives a label determined by the exclusive OR (XOR) of the labels of its incident vertices, subject to specified constraints. We first establish that the super subdivision of a path <img src=image/13441705_01.gif> by the complete bipartite graph <img src=image/13441705_02.gif> is discrete, demonstrating that replacing each edge of <img src=image/13441705_01.gif> with <img src=image/13441705_02.gif> preserves the discrete labeling property. Additionally, we prove that the square of a path <img src=image/13441705_01.gif>, obtained by connecting vertices at a distance of at most two, also admits a discrete labeling. Further, we examine the shadow graph <img src=image/13441705_04.gif>, constructed by duplicating <img src=image/13441705_01.gif> and connecting each vertex to its copy's neighbors, and show that it is also discrete. Another variant, the graph <img src=image/13441705_03.gif> formed by switching one pendant vertex in <img src=image/13441705_01.gif>, is similarly proven to support discrete labeling. Beyond paths, we explore star-related graphs, confirming that the shadow graph of a star <img src=image/13441705_05.gif> and the shadow graph of a bi-star <img src=image/13441705_06.gif> both permit discrete labeling. Extending the concept, we introduce strong discrete labeling, which imposes the stricter requirement by adding one more constraint to discrete labeling. Under this stronger condition, we prove that the star graph <img src=image/13441705_07.gif> admits a strong discrete labeling if and only if m is even, highlighting a parity-based constraint absent in standard discrete labeling. These findings expand the known families of graphs with discrete and strong discrete labeling, offering new insights into structural properties that facilitate such labeling. Potential applications include coding theory, network design, and algorithm optimization, while open questions remain regarding the extensibility of these results to other graph operations and more complex graph classes.

Assume that <em>G</em>(<em>V,E</em>) is a graph with <em>V</em> and <em>E</em> as its vertex and edge sets, respectively. We have <em>G</em> is simple, connected, and undirected. Given a function λ from a union of <em>V</em> and <em>E</em> into a set of <em>k</em>-integers from 1 until <em>k</em>. We call the function λ as a totally irregular total <em>k</em>-labeling if the set of weights of vertices and edges consists of different numbers. For any <em>u</em> ∈ <em>V</em>, we have a weight <em>wt</em>(<em>u</em>)=λ(<em>u</em>)+ ∑_{{uy ∈ E}} λ(<em>uy</em>). Also, it is defined a weight <em>wt</em>(<em>e</em>)= λ(<em>u</em>)+ λ(<em>uv</em>) + λ(<em>v</em>) for each <em>e</em>=<em>uv</em> ∈ <em>E</em>. A minimum <em>k</em> used in <em>k</em>-total labeling λ is named as a total irregularity strength of <em>G</em>, symbolized by ts(<em>G</em>). We discuss results on ts of some caterpillar graphs in this paper. The results are <em>ts</em>(S_{p,2,2,q}) = ⌈ (p+q-1)/2 ⌉ for <em>p</em>, <em>q</em> greater than or equal to 3, while ts(S_{p,2,2,2,p}) = ⌈(2p-1)/2 ⌉, <em>p</em> ≥ 4.

A reverse edge magic total (REMT) labeling for the graph Γ = (V(Γ), E(Γ)) along with cardinality of p = |V(G)| and q = |E(G)| respectively. A one-one map π: V(Γ) ∪ E(Γ) → {1, 2, … ,|V(Γ)|+|E(Γ)|} having a rule that for every edge, uv ϵ E(Γ), then π (uv) − {π (u) + π (v)} = c, where c is a magic number. A reverse edge magic total (REMT) labeling is called a reverse super edge magic total (RSEMT) labeling if vertex-set receive natural numbers. In present paper, we compute the RSEMT labeling on the forest of paths and subdivided star.

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.

An one-one correspondence function λ from V(G) ∪ E(G) to the set {1, 2, …, |V(G) | + |E(G) |} is a total labeling of a finite undirected graph G without loops and multiple edges, where |V(G) |and |E(G) | are the cardinality of vertex and edge set of G respectively. A perfectly antimagic total labeling is a totally antimagic total labeling whose vertex and edge-weights that are also pairwise distint. Perfectly antimagic total (PAT) graph is a graph having such labeling. The topic of discovering perfectly antimagic total labeling of some families of graphs is discussed in this paper. We also came up with certain conclusions about dual of a perfectly antimagic total graphs. Finally, we provided that the necessary and sufficient condition for a dual of a regular and irregular PAT graph to be a PAT graph.

(2,1)-Total labelling of outerplanar graphs

An edge-magic total labeling on a graph G is one-to-one map from V(G) ∪ E(G) onto the set of integers 1,2, ...,ν + e, where ν = |V(G)| and e = |E(G)|, with the property that, given any edge uv, f(u) + f(u, ν}) + f(ν) = k for every u,v ∈ V(G), and k is called magic valuation. An edge-magic total labeling f is called super edge-magic total if f(v(G)) = {1,2 ...,|V(G)|} and f(E(G)) = {|V(G)| + 1, |V(G)| + 2,... |V(G) + E(G)|}. In this paper we investigate edge-magic total labeling of a new graph called modified Watermill graph. Furthermore, the magic valuation of the modified Watermill graph WM(n) is , for n odd, n ≥ 3.

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.