Sum Of Labels Research Articles

Sum labelling graphs of maximum degree two

The concept of sum labelling was introduced in 1990 by Harary. A graph is a sum graph if its vertices can be labelled by distinct positive integers in such a way that two vertices are connected by an edge if and only if the sum of their labels is the label of another vertex in the graph. It is easy to see that every sum graph has at least one isolated vertex, and every graph can be made a sum graph by adding at most n2 isolated vertices to it. The minimum number of isolated vertices that need to be added to a graph to make it a sum graph is called the sum number of the graph.The sum number of several prominent graph classes (e.g., cycles, trees, complete graphs) is already well known. We examine the effect of taking the disjoint union of graphs on the sum number. In particular, we provide a complete characterisation of the sum number of graphs of maximum degree two, since every such graph is the disjoint union of paths and cycles.

The Space Complexity of Sum Labelling

A graph is called a sum graph if its vertices can be labelled by distinct positive integers such that there is an edge between two vertices if and only if the sum of their labels is the label of another vertex of the graph. Most papers on sum graphs consider combinatorial questions like the minimum number of isolated vertices that need to be added to a given graph to make it a sum graph. In this paper, we initiate the study of sum graphs from the viewpoint of computational complexity. Notice that every n-vertex sum graph can be represented by a sorted list of n positive integers where edge queries can be answered in mathscr {O}(log n) time. Therefore, upper-bounding the numbers used as vertex labels also upper-bounds the space complexity of storing the graph in the database. We show that every n-vertex, m-edge, d-degenerate graph can be made a sum graph by adding at most m isolated vertices to it, such that the largest numbers used as vertex labels grows as mathscr {O}(n^2d). This enables us to store the graph using mathscr {O}(mlog n) bits of memory. For sparse graphs (graphs with mathscr {O}(n) edges), this matches the trivial lower bound of Omega (nlog n). As planar graphs and forests have constant degeneracy, our result implies an upper bound of mathscr {O}(n^2) on their label numbers. The previously best known upper bound on the numbers needed for labelling general graphs with the minimum number of isolated vertices was mathscr {O}(4^n), due to Kratochvíl, Miller & Nguyen (2001). Furthermore, their proof was existential, whereas our labelling can be constructed in polynomial time.

Sum-distinguishing number of sparse hypergraphs

A vertex labeling of a hypergraph is sum distinguishing if it uses positive integers and the sums of labels taken over the distinct hyperedges are distinct. Let s(H) be the smallest integer N such that there is a sum-distinguishing labeling of H with each label at most N. The largest value of s(H) over all hypergraphs on n vertices and m hyperedges is denoted s(n,m). We prove that s(n,m) is almost-quadratic in m as long as m is not too large. More precisely, the following holds: If n≤m≤nO(1) then s(n,m)=m2w(m),where w(m) is a function that goes to infinity and is smaller than any polynomial in m.The parameter s(n,m) has close connections to several other graph and hypergraph functions, such as the irregularity strength of hypergraphs. Our result has several applications, notably:• We answer a question of Gyárfás et al. whether there are n-vertex hypergraphs with irregularity strength greater than 2n. In fact we show that there are n-vertex hypergraphs with irregularity strength at least n2−o(1).• In addition, our results imply that s∗(n)=n2/w(n) where s∗(n) is the distinguishing closed-neighborhood number, i.e., the smallest integer N such that any n-vertex graph allows for a vertex labeling with positive integers at most N so that the sums of labels on distinct closed neighborhoods of vertices are distinct.

On the hardness of determining the irregularity strength of graphs

Let G be a graph, and ℓ:E(G)→{1,…,k} be a k-labelling of G, i.e., an assignment of labels from {1,…,k} to the edges of G. We say that ℓ is irregular if no two distinct vertices of G are incident to the same sum of labels. The irregularity strength of G, denoted by s(G), is the smallest k such that irregular k-labellings of G exist. These notions were introduced in the late 1980s as an alternative way to deal with an optimisation problem where one aims at making a graph irregular by multiplying its edges in an optimal way. Since then, the irregularity strength has received a lot of attention, focusing mainly on proving bounds and investigating side aspects and variants.In this work, we consider the algorithmic complexity of determining the irregularity strength of a given graph. We prove that two close variants of this problem are NP-hard, which we suspect might indicate that the original problem is hard too. Namely, we prove that determining the distant irregularity strength, where only vertices within a certain distance are required to be incident to different sums of labels, and the multiset irregularity strength, where any two distinct vertices are required to be incident to different multisets of labels, are NP-hard problems.

On a List Variant of the Multiplicative 1-2-3 Conjecture

The 1-2-3 Conjecture asks whether almost all graphs can be (edge-)labelled with 1, 2, 3 so that no two adjacent vertices are incident to the same sum of labels. In the last decades, several aspects of this problem have been studied in literature, including more general versions and slight variations. Notable such variations include the List 1-2-3 Conjecture variant, in which edges must be assigned labels from dedicated lists of three labels, and the Multiplicative 1-2-3 Conjecture variant, in which labels 1, 2, 3 must be assigned to the edges so that adjacent vertices are incident to different products of labels. Several results obtained towards these two variants led to observe some behaviours that are distant from those of the original conjecture. In this work, we consider the list version of the Multiplicative 1-2-3 Conjecture, proposing the first study dedicated to this very problem. In particular, given any graph G, we wonder about the minimum k such that G can be labelled as desired when its edges must be assigned labels from dedicated lists of size k. Exploiting a relationship between our problem and the List 1-2-3 Conjecture, we provide upper bounds on k when G belongs to particular classes of graphs. We further improve some of these bounds through dedicated arguments.

On Square Sum Labeling of Two Families of Petersen Graphs

A labeling on a graph G with n vertices and m edges is called square sum if there exists a bijection f : V G ⟶ 0,1,2,3 , … , n − 1 such that the function f ∗ : E G ⟶ N defined by f ∗ s t = f s 2 + f t 2 , for all s t ∈ E G , is injective. A graph G having a square sum labeling is called square sum. In this study, we have investigated the square sum labeling of generalized Petersen graph and double generalized Petersen graph.

On Proper Labellings of Graphs with Minimum Label Sum

The 1–2–3 Conjecture, raised by Karoński, Łuczak and Thomason, states that almost every graph G admits a proper 3-labelling, i.e., a labelling of the edges with 1, 2, 3 such that no two adjacent vertices are incident to the same sum of labels. Another interpretation of this conjecture, that may be attributed to Chartrand et al., is that almost every graph G can be turned into a locally irregular multigraph M, i.e., with no two adjacent vertices having the same degree, by replacing each of its edges by at most three parallel edges. In other words, for almost every graph G there should be a locally irregular multigraph M with the same adjacencies and having a relatively small number of edges. The 1–2–3 Conjecture, if true, would indeed imply that there is such an M with \(|E(M)| \le 3|E(G)|\). In this work, we study proper labellings of graphs with the extra requirement that the sum of assigned labels must be as small as possible. In other words, given a graph G, we are looking for a locally irregular multigraph \(M^*\) with the smallest number of edges possible that can be obtained from G by multiplying edges. This problem is actually quite different from the 1–2–3 Conjecture, as we prove that there is no absolute constant k such that \(M^*\) can always be obtained from G by replacing each edge with at most k parallel edges. We investigate several aspects of this problem, covering algorithmic and combinatorial aspects. In particular, we prove that the problem of designing proper labellings with minimum label sum is \({\mathcal {N}}{\mathcal {P}}\)-hard in general, but solvable in polynomial time for graphs with bounded treewidth. We also conjecture that for almost every connected graph G there should be a proper labelling with label sum at most 2|E(G)|, which we verify for several classes of graphs.

On the total irregularity strength of convex polytope graphs

A vertex (edge) irregular total k-labeling ? of a graph G is a labeling of the vertices and edges of G with labels from the set {1,2,...,k} in such a way that any two different vertices (edges) have distinct weights. Here, the weight of a vertex x in G is the sum of the label of x and the labels of all edges incident with the vertex x, whereas the weight of an edge is the sum of label of the edge and the vertices incident to that edge. The minimum k for which the graph G has a vertex (edge) irregular total k-labeling is called the total vertex (edge) irregularity strength of G. In this paper, we are dealing with infinite classes of convex polytopes generated by prism graph and antiprism graph. We have determined the exact value of their total vertex irregularity strength and total edge irregularity strength.

On the role of 3s for the 1-2-3 Conjecture

The 1-2-3 Conjecture states that every connected graph different from K2 admits a proper 3-(edge-)labelling, i.e., can have its edges labelled with 1,2,3 so that no two adjacent vertices are incident to the same sum of labels. In connection with some recent optimisation variants of this conjecture, in this paper we investigate the role of the label 3 in proper 3-labellings of graphs. An intuition from previous investigations is that, in general, it should always be possible to produce proper 3-labellings assigning label 3 to a only few edges.We prove that, for every p≥0, there are various graphs needing at least p 3s in their proper 3-labellings. Actually, deciding whether a given graph can be properly 3-labelled with p 3s is NP-complete for every p≥0. We also focus on classes of 3-chromatic graphs. For various classes of such graphs (cacti, cubic graphs, triangle-free planar graphs, etc.), we prove that there is no p≥1 such that all their graphs admit proper 3-labellings assigning label 3 to at most p edges. In such cases, we provide lower and upper bounds on the number of 3s needed.

Group distance magic and antimagic hypercubes

Let G=(V,E) be a graph. A distance magic labeling of G is a bijective assignment of labels from {1,2,…,|V(G)|} to the vertices of G such that the sum of labels on neighbors of u is the same for all vertices u. It is known that the n-dimensional hypercube Qn has a distance magic labeling if and only if n≡2(mod4).Let Γ be an Abelian group of order |V(G)|. Analogously, a Γ-distance magic labeling of G is a bijection ℓ:V→Γ for which the sum of labels on neighbors of u is the same for all vertices u.In this paper we fully characterize Γ-distance magic labellings of n-dimensional hypercubes Qn. Namely we prove that for n odd, there does not exist a Γ-distance magic labeling of Qn for any Abelian group Γ of order |V(Qn)|, whereas for n even there exists a Γ-distance magic labeling of Qn for every Abelian group Γ of order |V(Qn)|.Similarly distance antimagic and Γ-distance antimagic labellings can be defined, where one aims to find a bijection such that the sums of labels are pairwise distinct for all the vertices. We study this problem and show in particular that there exists a Γ-distance antimagic labeling of Qn for any Abelian group Γ of order 2n if n is odd. We also indicate some relationships between Γ-closed distance magic and antimagic labellings and Γ-distance antimagic labellings.The proofs rely mostly on linear algebra.

Triple Equal Sum Labeling of Some Graphs

Triple Equal Sum Labeling of Some Graphs

Edge pair sum labeling of some classes of graphs

In 2013, we [3] introduced the concept of edge pair sum labeling and further studied in [5-15]. In this paper, we find some new results on edge pair sum labeling.

Some Labelings On Cycle With Parallel P4 Chord

In this paper we focused, to obtain some results on labeling of cycle graph, Cycle C2m (m ≥ 3) and C2m+1 (m ≥ 3) with parallel (path) P4 chords. We have proved, every cycle C2m (m ≥ 3) and C2m+1 (m ≥ 3) with parallel (path) P4 chords is a vertex odd mean graph and vertex even mean graph, though it satisfied their labeling. And also graph is proved for Square Sum labeling and Square Difference Labeling on cycle C2m (m ≥ 3) and C2m+1 (m ≥ 3) with parallel (path) P4 chords.

Further results on an equitable 1-2-3 Conjecture

In this work, we consider equitable proper labellings of graphs, which were recently introduced by Baudon, Pilśniak, Przybyło, Senhaji, Sopena, and Woźniak. Given a graph G, the goal is to assign labels to the edges so that (1) no two adjacent vertices are incident to the same sum of labels, and (2) every two labels are assigned about the same number of times. Particularly, we aim at designing such equitable proper k-labellings of G with k being as small as possible. In connection with the so-called 1-2-3 Conjecture, it might be that labels 1,2,3 are, a few obvious exceptions apart, always sufficient to achieve this just as in the non-equitable version of the problem.We provide results regarding some open questions about equitable proper labellings. Via a hardness result, we first prove that there exist infinitely many graphs for which more labels are required in the equitable version than in the non-equitable version. This remains true in the bipartite case. We finally show that, for every k≥3, every k-regular bipartite graph admits an equitable proper k-labelling.

Lucky and Proper Lucky Labeling of Quadrilateral Snake Graphs

The labeling is said to be lucky labeling of the graph if the vertices of the graph are labeled by natural number with satisfying the condition that sum of labels over the adjacent of the vertices in the graph are not equal and if vertices are isolated vertex then the sum of the vertex is zero. The least natural number which labelled the graph is the lucky number. The Lucky Number of graph G is denoted by η(G). The labeling defined as proper labeling if the vertices of the graph are labeled by natural number with fulfilling the condition that label of adjacent vertices is not the same. The labeling is defined as proper lucky labeling if labeling is proper and also lucky. The proper lucky number of graph G is denoted by η p (G). Here we obtain a lucky number and proper lucky number for family of quadrilateral snake graph such as quadrilateral snake graph, double quadrilateral snake graph, alternate quadrilateral snake graph and double alternate quadrilateral snake graph.

English

English

Sum number of fans

Graph labeling deals with assigning labels to one or more elements of a graph. It has a wide variety of applications including: coding theory, communication network addressing, data base management system and secret sharing schemes to mention a view. A mapping [Formula: see text] is called a sum labeling of a graph [Formula: see text] if it is an injection from [Formula: see text] to a set of positive integers, such that [Formula: see text] if and only if there exists a vertex [Formula: see text] such that [Formula: see text]. In this case, [Formula: see text] is called a working vertex. In general, a graph [Formula: see text] will require some isolated vertices to be labeled in this way. The least possible number of such isolated vertices is called the sum number of [Formula: see text]; denoted by [Formula: see text]. A sum labeling of a graph [Formula: see text] is said to be optimum if it labels [Formula: see text] by using [Formula: see text] isolated vertices. In this paper, we investigate the lower bounds for the number of isolates required by an even fan and an odd fan, and then we construct optimum sum labelling for the graphs to prove: [Formula: see text]

An Injective Version of the 1-2-3 Conjecture

In this work, we introduce and study a new graph labelling problem standing as a combination of the 1-2-3 Conjecture and injective colouring of graphs, which also finds connections with the notion of graph irregularity. In this problem, the goal, given a graph G, is to label the edges of G so that every two vertices having a common neighbour get incident to different sums of labels. We are interested in the minimum k such that G admits such a k-labelling. We suspect that almost all graphs G can be labelled this way using labels $$1,\dots ,\Delta (G)$$ . Towards this speculation, we provide bounds on the maximum label value needed in general. In particular, we prove that using labels $$1,\dots ,\Delta (G)$$ is indeed sufficient when G is a tree, a particular cactus, or when its injective chromatic number $$\mathrm{\chi _{\mathrm{i}}}(G)$$ is equal to $$\Delta (G)$$ .

Open Problems on Integral Sum Labellings

Let be a simple graph and be the set of all integers. An integral sum graph of a set of integers as the graph having as its vertex set, with two vertices adjacent whenever their sum is in A graph so obtained is called an integral sum graph. In other words, an integral sum graph of a finite subset is the graph with if and only if And is called an integral sum labelling of In the paper, many obtained conclusions are summarized and relevant open problem is raised.

Note on diagonal construction of Z2nm-supermagic labeling of Cn□Cm

A Γ-supermagic labeling of a graph G(V, E) with is a bijection from E to an Abelian group Γ of order k such that the sum of labels of all incident edges of every vertex is equal to the same element An existence of a -supermagic labeling of Cartesian product of two cycles, for n odd was proved recently. This along with an earlier result by Ivančo proved the existence of a -supermagic labeling of for every In this paper we present a simple unified labeling method for all