8,726 research outputs found
Radio Heronian Mean k-Graceful Labeling on Degree Splitting of Graphs
A mapping g:V\left(G\right)\rightarrow{k,k+1,\ldots,k+N-1} is a radio heronian mean k-labeling such that if for any two distinct vertices s and t of G, d\left(s,t\right)+\left\lceil\frac{g\left(s\right)+g\left(t\right)+\sqrt{g\left(s\right)g\left(t\right)}}{3}\right\rceil\geq1+D,for every s,t\in\ V(G), where D is the diameter of G. The radio heronian mean k-number of g, {rrhmn}_k(g), is the maximum number assigned to any vertex of G. The radio heronian mean number of G, {rhmn}_k(g), is the minimum value of {rhmn}_k(g) taken overall radio heronian mean labelings g of G. If {rhmn}_k(g)=\left|V\left(G\right)\right|+k-1, we call such graphs as radio heronian mean k-graceful graphs. In this paper, we investigate the radio heronian mean k-graceful labeling on degree splitting of graphs such as comb graph P_n\bigodot K_1, rooted tree graph {RT}_{n,n} hurdle graph {Hd}_n and twig graph\ {TW}_n.A mapping  is a radio heronian mean k-labeling such that if for any two distinct vertices  and  of , ,for every V(G), where  is the diameter of . The  radio heronian mean k-number of g, , is the maximum number assigned to any vertex of . The  radio heronian mean number of , , is the minimum value of  taken overall radio heronian mean labelings  of . If , we call such graphs as radio heronian mean k-graceful graphs. In this paper, we investigate the radio heronian mean k-graceful labeling on degree splitting of graphs such as comb graph , rooted tree graph  hurdle graph   and twig graph
On the graceful polynomials of a graph
Every graph can be associated with a family of homogeneous polynomials, one for every degree, having as many variables as the number of vertices.
These polynomials are related to graceful labellings: a graceful polynomial with all even coefficients is a basic tool, in some cases, for proving
that a graph is non-graceful, and for generating a possibly infinite class of non-graceful graphs. Graceful polynomials also seem interesting in their
own right. In this paper we classify graphs whose graceful polynomial has all even coefficients, for small degrees up to 4. We also obtain some
new examples of non-graceful graphs
On the Graceful Game
A graceful labeling of a graph with edges consists of labeling the
vertices of with distinct integers from to such that, when each
edge is assigned as induced label the absolute difference of the labels of its
endpoints, all induced edge labels are distinct. Rosa established two well
known conjectures: all trees are graceful (1966) and all triangular cacti are
graceful (1988). In order to contribute to both conjectures we study graceful
labelings in the context of graph games. The Graceful game was introduced by
Tuza in 2017 as a two-players game on a connected graph in which the players
Alice and Bob take turns labeling the vertices with distinct integers from 0 to
. Alice's goal is to gracefully label the graph as Bob's goal is to prevent
it from happening. In this work, we study winning strategies for Alice and Bob
in complete graphs, paths, cycles, complete bipartite graphs, caterpillars,
prisms, wheels, helms, webs, gear graphs, hypercubes and some powers of paths
Vertex Graceful Labeling-Some Path Related Graphs
Treating subjects as vertex graceful graphs, vertex graceful labeling, caterpillar, actinia graphs, Smarandachely vertex m-labeling
Decomposition of Certain Complete Graphs and Complete Multipartite Graphs into Almost-bipartite Graphs and Bipartite Graphs
In his classical paper [14], Rosa introduced a hierarchical series of labelings called Ï, Ï, ÎČ and α labeling as a tool to settle Ringelâs Conjecture which states that if T is any tree with m edges then the complete graph K2m+1 can be decomposed into 2m + 1 copies of T . Inspired by the result of Rosa [14] many researchers significantly contributed to the theory of graph decomposition using graph labeling. In this direction, in 2004, Blinco et al. [6] introduced Îł-labeling as a stronger version of Ï-labeling. A function g defined on the vertex set of a graph G with n edges is called a Îł-labeling if
(i) g is a Ï-labeling of G, (ii) G is a tripartite graph with vertex tripartition (A, B, C) with C = {c} and ÂŻb â B such that {ÂŻb, c} is the unique edge joining an element of B to c, (iii) g(a) \u3c g(v) for every edge {a, v} â E(G) where a â A, (iv) g(c) - g(ÂŻb) = n. Further, Blinco et al. [6] proved a significant result that the complete graph K2cn+1 can be cyclically decomposed into c(2cn + 1) copies of any Îł-labeled graph with n edges, where c is any positive integer. Recently, in 2013, Anita Pasotti [4] introduced a generalisation of graceful labeling called d-divisible graceful labeling as a tool to obtain cyclic G-decompositions in complete multipartite graphs. Let G be a graph of size e = d . m. A d-divisible graceful labeling of the graph G is an injective function g : V (G) â {0, 1, 2, . . . , d(m + 1) - 1} such that {|g(u) - g(v)|/{u, v} â E(G)} = {1, 2, . . . , d(m + 1) - 1}\{m + 1, 2(m + 1), . . . , (d - 1)(m + 1)}. A d-divisible graceful labeling of a bipartite graph G is called as a d-divisible α-labeling of G if the maximum value of one of the two bipartite sets is less than the minimum value of the other one. Further, Anita Pasotti [4] proved a significant result that the complete multipartite graph K (e/d +1)Ă2dc can be cyclically decomposed into copies of d-divisible α-labeled graph G, where e is the size of the graph G and c is any positive integer (K (e/d +1)Ă2dc contains e/d + 1 parts each of size 2dc). Motivated by the results of Blinco et al. [6] and Anita Pasotti [4], in this paper we prove the following results. i) For t â„ 2, disjoint union of t copies of the complete bipartite graph Km,n, where mâ„ 3, n â„ 4 plus an edge admits Îł-labeling.
ii) For t ℠2, t-levels shadow graph of the path Pdn+1 admits d-divisible α-labeling for any admissible d and n ℠1. Further, we discuss related open problems
On d-graceful labelings
In this paper we introduce a generalization of the well known concept of a
graceful labeling. Given a graph G with e=dm edges, we call d-graceful labeling
of G an injective function from V(G) to the set {0,1,2,..., d(m+1)-1} such that
{|f(x)-f(y)| | [x,y]\in E(G)}
={1,2,3,...,d(m+1)-1}-{m+1,2(m+1),...,(d-1)(m+1)}. In the case of d=1 and of
d=e we find the classical notion of a graceful labeling and of an odd graceful
labeling, respectively. Also, we call d-graceful \alpha-labeling of a bipartite
graph G a d-graceful labeling of G with the property that its maximum value on
one of the two bipartite sets does not reach its minimum value on the other
one. We show that these new concepts allow to obtain certain cyclic graph
decompositions. We investigate the existence of d-graceful \alpha-labelings for
several classes of bipartite graphs, completely solving the problem for paths
and stars and giving partial results about cycles of even length and ladders.Comment: In press on Ars Combi
An evaluation on the gracefulness and colouring of graphs
In this thesis we shall introduce two interesting topics from graph theory and begin to explore what happens when we combine these together. We will be focusing on an area known as graph colouring and assessing it alongside a very unique set of graphs called graceful graphs. The two topic areas, although not mixed together often, nicely support each other in introducing various findings from each of the topics. We will start by investigating graceful graphs and determining what classes of graph can be deemed to be graceful, before introducing some of the fundamentals of graph colouring. Following this we can then begin to investigate the two topics combined and will see a whole range of results, including some fascinating less well known discoveries. Furthermore, we will introduce some different types of graph colouring based off the properties of graceful graphs. Later in the thesis there will also be a focus on tree graphs, as they have had a huge influence on research involving graceful graphs over the years. We will then conclude by investigating some results that have been formulated by combining graceful graphs with a type of graph colouring known as total colouring
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