Total edge irregularity strength of complete graphs and complete bipartite graphs

AbstractA total edge irregular k-labelling ν of a graph G is a labelling of the vertices and edges of G with labels from the set {1,…,k} in such a way that for any two different edges e and f their weights φ(f) and φ(e) are distinct. Here, the weight of an edge g=uv is φ(g)=ν(g)+ν(u)+ν(v), i. e. the sum of the label of g and the labels of vertices u and v. The minimum k for which the graph G has an edge irregular total k-labelling is called the total edge irregularity strength of G.We have determined the exact value of the total edge irregularity strength of complete graphs and complete bipartite graphs

Minimum-Weight Edge Discriminator in Hypergraphs

In this paper we introduce the concept of minimum-weight edge-discriminators in hypergraphs, and study its various properties. For a hypergraph $\mathcal H=(\mathcal V, \mathcal E)$, a function $\lambda: \mathcal V\rightarrow \mathbb Z^{+}\cup\{0\}$ is said to be an {\it edge-discriminator} on $\mathcal H$ if $\sum_{v\in E_i}{\lambda(v)}>0$, for all hyperedges $E_i\in \mathcal E$, and $\sum_{v\in E_i}{\lambda(v)}\ne \sum_{v\in E_j}{\lambda(v)}$, for every two distinct hyperedges $E_i, E_j \in \mathcal E$. An {\it optimal edge-discriminator} on $\mathcal H$, to be denoted by $\lambda_\mathcal H$, is an edge-discriminator on $\mathcal H$ satisfying $\sum_{v\in \mathcal V}\lambda_\mathcal H (v)=\min_\lambda\sum_{v\in \mathcal V}{\lambda(v)}$, where the minimum is taken over all edge-discriminators on $\mathcal H$. We prove that any hypergraph $\mathcal H=(\mathcal V, \mathcal E)$, with $|\mathcal E|=n$, satisfies $\sum_{v\in \mathcal V} \lambda_\mathcal H(v)\leq n(n+1)/2$, and equality holds if and only if the elements of $\mathcal E$ are mutually disjoint. For $r$-uniform hypergraphs $\mathcal H=(\mathcal V, \mathcal E)$, it follows from results on Sidon sequences that $\sum_{v\in \mathcal V}\lambda_{\mathcal H}(v)\leq |\mathcal V|^{r+1}+o(|\mathcal V|^{r+1})$, and the bound is attained up to a constant factor by the complete $r$-uniform hypergraph. Next, we construct optimal edge-discriminators for some special hypergraphs, which include paths, cycles, and complete $r$-partite hypergraphs. Finally, we show that no optimal edge-discriminator on any hypergraph $\mathcal H=(\mathcal V, \mathcal E)$, with $|\mathcal E|=n (\geq 3)$, satisfies $\sum_{v\in \mathcal V} \lambda_\mathcal H (v)=n(n+1)/2-1$, which, in turn, raises many other interesting combinatorial questions.Comment: 22 pages, 5 figure

THE TOTAL IRREGULARITY STRENGTH OF SOME COMPLETE BIPARTITE GRAPHS

This paper deals with the total irregularity strength of complete bipartite graph  where  and . &nbsp

On H-irregularity Strengths of G-amalgamation of Graphs

A simple graph G=(V(G),E(G)) admits an H-covering if every edge in E(G) belongs at least to one subgraph of G isomorphic to a given graph H. Then the graph G admitting H-covering admits an H-irregular total k-labeling f: V(G) U E(G) \to {1, 2, ..., k} if for every two different subgraphs H\u27 and H\u27\u27 isomorphic to H there is $wt_{f}(H\u27) \neq wt_{f}(H\u27\u27)$, where $wt_{f}(H)= \sum \limits_{v\in V(H)} f(v) + \sum \limits_{e \in E(H)} f(e)$ is the associated H-weight. The minimum k for which the graph G has an H-irregular total k-labeling is called the total H-irregularity strength of the graph G.In this paper, we obtain the precise value of the total H-irregularity strength of G-amalgamation of graphs

Further Results on (a, d) -total Edge Irregularity Strength of Graphs

ليكن  رسمًا بيانيًا بسيطًا على رؤوس l وحواف m مع إجمالي h -  وضع العلامات  . فان   تسمى (ا,د)- وسم غير منتظم للحافة الإجمالية إذا وجد تطابق متقابل وليكن   معرفة بواسطة   لكل   , حيث  . كذلك قيمة  يقال لها وزن الحافة . يشار الى (ا,د)-اجمالي قوة عدم انتظام الحواف للرسم البياني G ب  وهي اقل h التي يقبلها G   للحافة -(ا,د) الغير منتظمة للعلامة-h . في هذه المقالة تم فحص,  لبعض عائلات الرسم البياني الشائعة. بالاضافة الى ذلك تم حل المسالة المفتوحة  بشكل ايجابي. م تسمى ρ (أ ، د) - وسم غير منتظم للحافة الإجمالية إذا كان هناك تطابق واحد لواحد ، قل ψ: E (G) → {a ، a + d ، a + 2d ،… + a + (m- 1) د} محدد بواسطة ψ (uv) = ρ (u) + ρ (v) + ρ (uv) لجميع uv∈E (G) ، حيث a≥3 ، d≥2. أيضًا ، يُقال إن القيمة ψ (uv) هي وزن حافة الأشعة فوق البنفسجية. يشار إلى قوة عدم انتظام الحافة الإجمالية (أ ، د) للرسم البياني G بواسطة (a ، d) -tes (G) وهي أقل h التي يقبلها G (أ ، د) - علامة h غير منتظمة للحافة. في هذه المقالة ، يتم فحص (أ ، د) -tes (G) لبعض عائلات الرسم البياني الشائعة. بالإضافة إلى ذلك ، يتم حل المشكلة المفتوحة (3،2) - tes (K_ (m ، n)) ، m ، n> 2 بشكل إيجابي.Consider a simple graph   on vertices and edges together with a total  labeling . Then ρ is called total edge irregular labeling if there exists a one-to-one correspondence, say  defined by  for all  where  Also, the value  is said to be the edge weight of . The total edge irregularity strength of the graph G is indicated by  and is the least  for which G admits   edge irregular h-labeling.  In this article,   for some common graph families are examined. In addition, an open problem is solved affirmatively

The 1-2-3 Conjecture for Hypergraphs

A weighting of the edges of a hypergraph is called vertex-coloring if the weighted degrees of the vertices yield a proper coloring of the graph, i.e., every edge contains at least two vertices with different weighted degrees. In this paper we show that such a weighting is possible from the weight set {1,2,...,r+1} for all hypergraphs with maximum edge size r>3 and not containing edges solely consisting of identical vertices. The number r+1 is best possible for this statement. Further, the weight set {1,2,3,4,5} is sufficient for all hypergraphs with maximum edge size 3, up to some trivial exceptions.Comment: 12 page