4,737 research outputs found
On the neighbour sum distinguishing index of planar graphs
Let be a proper edge colouring of a graph with integers
. Then , while by Vizing's theorem, no more than
is necessary for constructing such . On the course of
investigating irregularities in graphs, it has been moreover conjectured that
only slightly larger , i.e., enables enforcing additional
strong feature of , namely that it attributes distinct sums of incident
colours to adjacent vertices in if only this graph has no isolated edges
and is not isomorphic to . We prove the conjecture is valid for planar
graphs of sufficiently large maximum degree. In fact even stronger statement
holds, as the necessary number of colours stemming from the result of Vizing is
proved to be sufficient for this family of graphs. Specifically, our main
result states that every planar graph of maximum degree at least which
contains no isolated edges admits a proper edge colouring
such that for every edge of .Comment: 22 page
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 , a function is said to be an {\it edge-discriminator} on if
, for all hyperedges , and
, for every two
distinct hyperedges . An {\it optimal
edge-discriminator} on , to be denoted by , is
an edge-discriminator on satisfying , where
the minimum is taken over all edge-discriminators on . We prove
that any hypergraph , with , satisfies ,
and equality holds if and only if the elements of are mutually
disjoint. For -uniform hypergraphs , it
follows from results on Sidon sequences that , and
the bound is attained up to a constant factor by the complete -uniform
hypergraph. Next, we construct optimal edge-discriminators for some special
hypergraphs, which include paths, cycles, and complete -partite hypergraphs.
Finally, we show that no optimal edge-discriminator on any hypergraph , with , satisfies
, which, in turn,
raises many other interesting combinatorial questions.Comment: 22 pages, 5 figure
On Total Irregularity Strength of Double-Star and Related Graphs
AbstractLet G = (V, E) be a simple and undirected graph with a vertex set V and an edge set E. A totally irregular total k-labeling f : V ∪ E → {1, 2,. . ., k} is a labeling of vertices and edges of G in such a way that for any two different vertices x and x1, their weights and are distinct, and for any two different edges xy and x1y1 their weights f (x) + f (xy) + f (y) and f (x1) + f (x1y1) + f (y1) are also distinct. A total irregularity strength of graph G, denoted byts(G), is defined as the minimum k for which G has a totally irregular total k-labeling. In this paper, we determine the exact value of the total irregularity strength for double-star S n,m, n, m ≥ 3 and graph related to it, that is a caterpillar S n,2,n, n ≥ 3. The results are and ts(S n,2,n) = n
On the Total Irregularity Strength of Fan, Wheel, Triangular Book, and Friendship Graphs
AbstractA totally irregular total k-labeling λ: V ∪ E → {1, 2, · · ·, k} of a graph G is a total labeling such that G has a total edge irregular labeling and a total vertex irregular labeling at the same time. The minimum k for which a graph G has a totally irregular total k-labeling is called the total irregularity strength of G, denoted by ts(G). In this paper, we investigate some graphs whose total irregularity strength equals to the lower bound
Inverse problem for Albertson irregularity index
Graph indices have attracted great interest as they give us numerical clues for several properties of molecules. Some indices give valuable information on the molecules under consideration using mathematical calculations only. For these reasons, the calculation and properties of graph indices have been in the center of research. Naturally, the values taken by a graph index is an important problem called the inverse problem. It requires knowledge about the existence of a graph having index equal to a given number. The inverse problem is studied here for Albertson irregularity index as a part of investigation on irregularity indices. A class of graphs is constructed to Show that the Albertson index takes all positive even integers. It has been proven that there exists at least one tree with Albertson index equal to every even positive integer but 4. The existence of a unicyclic graph with irregularity index equal to m is shown for every even positive integer m except 4. It is also shown that the Albertson index of a cyclic graph can attain any even positive integer.Publisher's Versio
Inverse problem for Albertson irregularity index
Graph indices have attracted great interest as they give us numerical clues for several properties of molecules. Some indices give valuable information on the molecules under consideration using mathematical calculations only. For these reasons, the calculation and properties of graph indices have been in the center of research. Naturally, the values taken by a graph index is an important problem called the inverse problem. It requires knowledge about the existence of a graph having index equal to a given number. The inverse problem is studied here for Albertson irregularity index as a part of investigation on irregularity indices. A class of graphs is constructed to Show that the Albertson index takes all positive even integers. It has been proven that there exists at least one tree with Albertson index equal to every even positive integer but 4. The existence of a unicyclic graph with irregularity index equal to m is shown for every even positive integer m except 4. It is also shown that the Albertson index of a cyclic graph can attain any even positive integer.Publisher's Versio
- …