On Harmonious Graphs

Several of the most studied graph labelings are injective functions, this constraint precludes some graphs from admitting such labelings; a well-known example is given by the family of trees that cannot be harmoniously labeled. In order to study the existence of these labelings for certain graphs, the injectivity constraint is often dropped. In this work we eliminate this condition for two different, but related, additive vertex labelings such as the harmonious and arithmetic labelings. The new labelings are called semi harmonious and semi arithmetic. We consider some families of graphs that do not admit the injective versions of these labelings, among the graphs considered here we have cycles and other cycle-related graphs, including the analysis of some operations like the Cartesian product and the vertex or edge amalgamation; in addition, we prove that all trees admit a semi harmonious labeling. Something similar is done with the concept of arithmetic labeling, studying finite unions of semi arithmetic graphs together with some general results.

Research in graph theory has lead to one of the important area called labeling of graphs. There are different types of labeling such as graceful labeling, magic labeling, edge-graceful labeling, prime labeling, radio labeling, harmonious labeling etc., In this paper, we investigate the existence of edge graceful labeling for the Corona graph Cn⊙K2¯ (n is odd n ≥ 3). Also we have defined a flower graph which does not admit an edge-graceful labeling.

Labelling problems are very useful in modelling, solving problems in other areas and enhance the study of algebraic structures. Nowadays, a more practical, accurate and efficient way of labeling a graph is using some mathematical methods like integer programming or constraint programming. This will improve the scope and quality of research. Here, considering the applications of Graceful and Harmonious labelings, we introduce a few new types of labeling, namely, gracefully harmonious labeling, gracefully felicitous labeling, gracefully elegant labeling, gracefully strongly c-elegant labeling and gracefully semi-harmonious labeling. In this paper, we prove that some complete multi-partite graphs is gracefully harmonious, gracefully felicitous and gracefully semi-harmonious. Keywords: Labeling; multi-partite graph; gracefully harmonious; gracefully felicitous; gracefully semi-harmonious. 2010 Mathematics Subject Classification: 05C78

Cordial labeling of hypertrees

Although Min-CDS in general graphs is hard to approximate, the restriction to certain special graph classes admits much better approximation results. Min-CDS in planar graphs remains NP-hard even for planar graphs that are regular of degree 4 [57]. The related problem, Min-DS in planar graphs, is also NP-hard even for planar graphs with maximum vertex degree 3 and planar graphs that are regular of degree 4 [57]. It is well known that Min-DS in planar graphs possesses a polynomial-time approximation scheme (PTAS) based on the shifting strategy [3]: For any constant e >0, there is a polynomial-time 1+e-approximation algorithm. Thus, it is immediate to conclude that Min-CDS in planar graphs can be approximated within a factor 3+e for any e>0 in polynomial time. However, the degree of the polynomial grows with 1∕e and hence, the approximation scheme is hardly practical.

Minimal multicut and maximal integer multiflow: A survey

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we verify the total coloring conjecture for every 1-planar graph G if either Δ(G) ≥ 9 and g(G) ≥ 4, or Δ(G) ≥ 7 and g(G) ≥ 5, where Δ(G) is the maximum degree of G and g(G) is the girth of G.

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.

This study investigates the (a,d)-face antimagic labeling on three classes of planar graphs, namely the house graph, the double house graph, and the triangular ladder graph. An (a,d)-face antimagic labeling is defined as a bijective function from the set of vertices, edges, and faces to the set of positive integers, such that the sum of the labels assigned to the vertices and edges incident with each face forms an arithmetic sequence. In this work, specific labeling constructions are developed for the vertices, edges, and faces of each graph to ensure that the face antimagic property is satisfied. Illustrative examples are presented to demonstrate the labeling schemes that produce face weights forming an arithmetic sequence with a fixed common difference of d = 1. The findings confirm that all three classes of graphs admit valid face antimagic labelings. This research contributes to the advancement of graph labeling theory, particularly in the study of structured planar graphs

One important problem in graph theory is graph coloring or graph labeling.Labeling problem is a well-studied problem due to its wide applications, especially in frequency assignment in(mobile) communication systems, coding theory, x-ray crystallography, radar, circuit design, etc.For two non-negative integers, labeling of a graph is a function from the node set to the set of non-negative integers.For any two distinct vertices and of , a radio labeling is an injective function : 0 such that , | | 1 where is the diameter of .The radio number of , is the maximum value of taken overall radio labelings of .This paper determines the radio number of double triangular snake graphs.

An $L(2,1)$-labeling of a graph is a mapping $c:V(G) \to \{0,\ldots,K\}$ such that the labels assigned to neighboring vertices differ by at least $2$ and the labels of vertices at distance two are different. Griggs and Yeh [SIAM J. Discrete Math. 5 (1992), 586―595] conjectured that every graph $G$ with maximum degree $\Delta$ has an $L(2,1)$-labeling with $K \leq \Delta^2$. We verify the conjecture for planar graphs with maximum degree $\Delta \neq 3$.

Gaussian anti magic labeling in some special graphs

Let G(V (G), E(G)) be a finite simple graph with |V (G)| = G and |E(G)| = eG. Let H be a subgraph of G. The graph G is said to be (a, d)-H-antimagic covering if every edge in G belongs to at least one of the subgraphs G isomorphic to H and there is a bijective function ξ : V ∪ E → {1, 2, …,νG + eG} such that all subgraphs H′ isomorphic to H, the H′-weightsw(H′)=∑v∈V(H′)ξ(v)+∑e∈E(H′)ξ(e)constitutes an arithmetic progression {a, a + d, a + 2d, …, a + (t − 1)d}, where a and d are positive integers and t is the number of subgraphs G isomorphic to H. Such a labeling is called super if the vertices contain the smallest possible labels. This research provides super (a, d)-C3-antimagic total labelng on triangular ladder TLn for n ≥ 2 and super (a, d)-Cs+2-antimagic total labeling on generalized Jahangir Jk,s for k ≥ 2 and s ≥ 2.

A single circular DNA or polypeptide strand can be used to construct nanostructures based on the mathematical concept of strong traces. It is known that if a special face graph of a 2-connected plane graph G is a tree, then G has a thickened graph F(G) with only one boundary component. However, to ensure that this thickened graph contains only a single boundary component, it is not necessary for the special face graph of G to be a tree. In this paper, we summarize a combinatorial method, called the flat-based operations, for obtaining a matrix M0 where det(M0) determines whether F(G) associated with G contains just one boundary component. By applying this method, we can easily determine if the number of boundary components of some F(G) is one. Further, we characterize a type of 2-connected plane graph whose thickened graph has more than one boundary component and then has no antiparallel strong trace.

This paper studies the strategy consensus problem of networked evolutionary agmes (NEGs) under degree-based pinning control. Firstly, two necessary and sufficient conditions are given for the strategy consensus of NEGs based on different premises. Secondly, an algorithm is obtained to get the minimal-agent control based on degree for making the NEGs achieve the strategy consensus. Thirdly, as application, we present two sufficient conditions for the strategy consensus of NEGs with two kinds of special networked graph: wheel graph and f<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf>× 3 flower graph. Finally, an illustrative example is employed to show the effective of the obtained results.