J. Comp. & Math. Sci. Vol.2 (2), 254-259 (2011)
A Note on Perception of Definitions in Graph Labellings of Trees A. KRISHNAA Department of Mathematics and Statistics Mohan Lal Sukhadia University, Udaipur (Rajasthan), India. email:godseeking1@yahoo.com ABSTRACT This note aims to shed light on the misconception of equating concept of definitions of graph labellings particularly those of trees with concept of definitions of other branches of mathematics like say dealing with numbers. This fundamental misconception has given rise to making some conjectures about labellings of trees which can not be proven but are believed to be true even without proofs. A labelling of a graph is assigning numbers to vertices such that the induced labels of the edges form a certain pattern. Let us take a graph G with p vertices and q edges. A harmonious labelling is a one-one mapping g:V(G)→{0,1,2,…,q-1} such that the induced mapping g*(e)=(g(x)+g(y))mod q for each edge xy is bijective (repetition of exactly one vertex label is allowed for trees). A felicitous labelling of G is a one-one mapping g:V(G)→{0,1,2,…,q} such that the induced mapping g*(e)=((g(x)+g(y))mod q is bijective. A sequential labelling of G is a one-one mapping g:V(G)→{0,1,2,…,q-1}such that the induced mapping*(e)=g(x)+g(y) is bijective. A graceful labelling is an injection g:V(G)→{0,1,2,…,q}such that the induced function given by g*(x,y)=|g(x)-g(y)| for all edges xy is injective. An antimagic labelling is one in which the edges are labeled with {1,2,3,…,q}such that the sum of the labels of all incident edges at a vertex for all the vertices are distinct.
INTRODUCTION All trees have the common feature of being acyclic and connected. Apart from that there is no limitation on shape, size and appearance or structure of the trees thus implying that infinite shapes of trees are
possible. Labelling of any kind depends on the structure or shape of the tree which determines the relation between adjacencies and incidences of vertices and edges and hence the effect on labelling of any kind. There seems to be a perception that definition of tree is like definition in some
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other branches of mathematics such that the infinite shapes of trees can be determined by its being connected and acyclic alone and thus predict the behaviour of the infinite varieties of trees for labelling of a certain kind. Graphs being pictorial, definition alone can not control or determine the shape and size of the tree and hence the existence of certain labelling as well, which depends on shape and size of the graph. Specialized results of existence of labeling of a certain kind can be applicable for certain kind(s) of tree but how can it be generalized to infinite kinds of trees since a graph is not a number or a fixed structure but a pictorial representation with no bounds on its shape, adjacencies and incidencies of its vertices and edges. In numbers for instance the definition of an even or odd number can be applied to any number imaginable however large but a graph is not a number or a fixed structure. It is a pictorial representation with infinite possibilities. Hence a graph tree being connected and acyclic alone can not be the basis of conjecture that all trees are graceful or of any other type. A graph with three vertices can also be a tree and a graph with fifty thousand vertices can also be a tree with edges and vertices connected in infinite number of ways. For a certain kind of labeling for the established classes of trees, the methods devised so far can be used to compute the vertex labels and for the unclassified trees either manually or all possible permutations and combinations of vertex labels can be examined by the specialized software developed in Krishnaa (2001). It is beyond the mind’s conception to even visualize the adjacencies and incidences of such large
graphs and that too of the infinite varieties of trees. The infinite shapes of trees can neither be visualized nor are likely to be classified barring a few. For instance, caterpillars, lobsters, symmetrical trees (a kind of rooted tree where every level contains vertices of the same degree), regular bamboo trees (a rooted tree consisting of branches of equal length the endpoints of which are identified with end points of stars of equal size), olive trees, spider, firecrackers, banana trees are certain kinds of trees. Of these spiders are known to be graceful in certain cases, symmetrical trees (a kind of rooted tree) are known to be graceful, rooted trees in certain cases are known to be graceful, firecrackers are known to be graceful and banana trees are known to be graceful in certain cases, caterpillars are known to be harmonious. Let us look at a case of the ever expanding ways of development of classifications and specialized results. The following algorithms namely Algorithm HS gives harmonious and sequential labellings and Algorithm G gives graceful labellings of a bipartite tree: Algorithm HS: 1. Draw the tree as a bipartite graph in two partite sets denoted as Left (L) and Right(R). Let the number of vertices in L be x. 2. Number the vertices in L starting from top going to bottom consecutively as 0,1,‌,(x-1). 3. Number the vertices in R starting from top going to bottom consecutively as (x-1),x,(x+1),‌,(q-1). (q is the no. of
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edges ).Note that these numbers are the vertex labels. 4. Computer the edge labels by adding them modulo q for harmonious and simple addition for sequential. 5. The resulting labelling is harmonious and sequential.
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Figure 1 : Sequential and harmonious bipartite trees;sequential and harmonious edge labels written on the edges on left and right respectively separated by a comma. Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)
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Algorithm G : 1. Draw the tree as a bipartite graph in two partite sets denoted as Left (L) and Right (R). Let the number of vertices in L be x. 2. Number the vertices in L starting from top going to bottom consecutively as 0,1,‌,(x-1).
Number the vertices in R starting from bottom going to top consecutively as x,(x+1),(x+2),‌,q (q is the no. of edges). Note that these numbers are the vertex labels. 4. Compute the edge labels by taking the absolute value of the difference of the incident vertex labels. 5. The resulting labelling is graceful. 10
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Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)
A. Krishnaa, J. Comp. & Math. Sci. Vol.2 (2), 254-259 (2011)
Figure 3 : T(10,9)
Figure 5 : T(10,9) The tree T(10,9) when drawn as rooted trees as in figures 3,4,5,6 does not give harmonious, sequential or graceful labelling with these algorithms because the adjacencies and incidencies of the vertices
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Figure 4 : T(10,9)
Figure 6 : T(10,9) and edges change completely even though the tree is still T(10,9) and still following the definition of being connected and acyclic. Now, the available results for rooted trees being that the symmetric trees (a rooted tree
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in which every level contains vertex of same degree) are graceful by Bermond and Sotteau (1976) and Sethuraman and Jesintha (three papers in 2008) come up with a new class and family of rooted trees and show that rooted trees in which every level contains pendant vertices and the degrees of the internal vertices in the same level are equal are graceful. Thus it is seen that the division and classification can go on endlessly with more and more specialized results obtained. This is so because of the great number of possibilities of drawing the same tree. As the order and size of the tree increase the possibilities of classifications become even more and thus eventually yielding more and more specialized results due to no limit on the shapes of drawing a tree (classification depends on the shape). Thus it should be noted that pictorial representation of graphs in particular trees gives rise to infinite number of trees which makes it impossible to classify them all and makes the tree definition distinct from definitions in some other branches of mathematics. This renders the tree definition infeasible to use in generalized results.
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REFERENCES 1.
Bermond J. C. and Sotteau D. (1976), Graph decompositions and G-design, proc. British Combin. Cong., Congr. Numer. : XV, 53-72 (1976).
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Cahit I., Status of graceful tree conjecture in 1989. In R. bodendick and R. R, Henn (Eds.), Topics in Combinatorics and Graph Theory, Physica-Verlag, Heidelberg, (1989). Gallian J. A., A dynamic survey of graph labeling, The Electronic Journal of Combinatorics, 5 (1998), #DS6, (1998). Koh K.M., Rogers D.G. and Tan T., A graceful arboretum: a survey of graceful trees, proceedings of FrancoSoutheast Asian Conf., Singapore, 278-287, (1979). Krishnaa A., Computer modelling of graph labelings, proceedings of the National Conference on Mathematical and Computational Models, 293-301, December 27-28, Coimbatore, India, Allied Publishers, India, (2001). Ringel G.A., Llado A.and Serra O., Another tree conjecture, Bull. Inst. Combin. Appl., 18, 83-85 (1996). Sethuraman G. and Jesintha J., Gracefulness of a family of rooted trees, Far East J. Appl. Math., 30, 143-159 (2008). Sethuraman G. and Jesintha J.,A new family of graceful rooted trees, proc. National Conf. Appl. Math., 74-80, (2008). Sethuraman G. and Jesintha J., A new class of graceful rooted trees, J. Discrete Math. Sci. Crypt., 11, 421435, (2008).
Journal of Computer and Mathematical Sciences Vol. 2, Issue 2, 30 April, 2011 Pages (170-398)