G. Sethuraman

A new class of graceful rooted trees

Abstract In [1], Bermond and Sotteau have shown that all rooted trees in which all the vertices at the same distance from the root have the same degree (symmetrical trees) are graceful. In this paper we extend this result. More precisely, we prove that all rooted trees, in which every level contains pendant vertices and the degrees of internal vertices within the same level are equal but the degree of internal vertices in the different levels could be different, are graceful.

All Arbitrarily Fixed Generalized Banana Trees Are Graceful

Consider a set of caterpillars, having equal and fixed diameter, in which one of the penultimate vertices is of arbitrary degree and all the other internal vertices including the other penultimate vertex are of fixed even degree. Merge an end-vertex adjacent to the penultimate vertex of fixed even degree of each of such caterpillars together. The rooted tree thus obtained is called Arbitrarily Fixed Generalized Banana Tree. In this paper we prove that all arbitrarily fixed generalized banana trees are graceful. This would imply that “all banana trees are graceful” and “all generalized banana trees are graceful” as corollaries.

Gracefulness of union of cycle with parallel chords and complete bipartite graphs or paths

Abstract A graph denoted Hn , is called a cycle with parallel chords if Hn is obtained from the cycle Cn : u 1, u 2,…, un (n ≥ 6) by adding the chords u 2 un , u 3 u n−1 ,…, uαuβ , where and if n is odd or if n is even (see Figure 1). In this paper we prove the following results. (1) For all p, q ≥ 1 and for all odd positive integers n ≥ 5, Hn ∪ Kp,q is graceful, where Kp,q is a complete bipartite graph. (2) For n ≥ 6, Hn ∪ Sm is graceful, where m ≥ 1 when n is odd and while when n is even, or . Here Sm denotes a star of size m. (3) For n ≥ 6, Hn ∪ Pm is graceful, where m = n or n − 2 depends on n = 1 or 3(mod 4) or m = n − 1 or n − 3 depends on n = 0 or 2 (mod 4). Here Pm denotes a path of order m.

Any Tree with m edges can be embedded in a Graceful Tree with less than 4m edges and in a graceful planar graph

Abstract A function f is called a graceful labeling of a graph G with m edges, if f is an injective function from V ( G ) to { 0 , 1 , 2 , … , m } such that when every edge u v is assigned the edge label | f ( u ) − f ( v ) | , then the resulting edge labels are distinct. A graph which admits a graceful labeling is called a graceful graph. In this paper, we prove a basic structural property of graceful graphs, that every tree can be embedded as a spanning subtree in a graceful planar graph. Also we show that any tree with m edges can be embedded in a graceful tree with less than 4 m edges. A range-relaxed graceful labeling f is defined from V ( G ) to 0 , 1 , 2 , … , m ′ , where m ′ ≥ m . We improve the bound 2 m − d i a m ( T ) on the range-relaxed graceful labeling given by Van Bussel (2002) for a tree T .

(2,1)-Total labeling of cycle with parallel paths

Abstract A (p, 1)-total labeling of a graph G is an assignment of integers to V (G) ∪ E(G) such that (i) any two adjacent vertices of G receive distinct integers, (ii) any two adjacent edges of G receive distinct integers, and (iii) a vertex and an edge incident receive integers that differ by at least p in absolute value. The span of a (p, 1)-total labeling is the maximum difference between two labels. The minimum of span of all possible (p, 1)-total labeling of G is called the (p, 1)-total number and denoted by λp (G). The well known Havet and Yu Conjecture [6] states that for any connected graph G with ∆(G) ≤ 3 and G 6= K4, λ T 2 (G) ≤ 5. In this paper, we determine the (2, 1)-total number of cycle with parallel paths. This result supports the Havet and Yu conjecture.

Gracefulness of graphs obtained from vertex duplication

A graceful labeling of a graph G with n edges is an injection f : V (G) → {0, 1, 2, . . . , n} with the property that the resulting edge labels are distinct where an edge incident with vertices u and v is assigned the label |f(u) − f(v)|. The characterization of graceful graphs is one of most difficult problems in graph theory. In this paper, we study the effectiveness of the graph operation, the duplication of all the vertices of graphs in obtaining graceful graphs. More precisely, we prove the following results. 1. If G admits a stronger version of graceful labeling, α-labeling, then the graph D(G) obtained by duplication of all the vertices of G admits α-labeling. 2. The graphs obtained by the duplication of all the vertices of the non-graceful graphs, C4∪K1,n, for all n ≥ 1 and n ̸= 2 and C3∪Pn, for all n ≥ 1 are graceful. Mathematics Subject Classification: 05C78, 05C76

Graceful and Cordial Labeling Of Subdivision Of Graphs

Abstract An edge uv is said to be subdivided if the edge uv is replaced by the path P : u w v , where w is the new vertex. A graph obtained by subdividing each edge of a graph G is called subdivision of the graph G, and is denoted by S ( G ) . A shell graph of size n ≥ 4 , denoted C ( n , n − 3 ) is the graph obtained from the cycle C n ( v 0 , v 1 , v 2 , ⋯ , v n − 1 ) by adding n − 3 consecutive chords incident with a common vertex v 0 (say) called apex of the shell graph. In this paper, we show that the graph S ( K 2 , n ) is graceful and cordial, for n ≥ 1 and the graph S ( C ( n , n − 3 ) ) is graceful and cordial for n ≥ 4 .

Generating Graceful Trees from Caterpillars by Recursive Attachment

Abstract A graceful labeling of a graph G with n edges is an injection f : V ( G ) → { 0 , 1 , 2 , ⋯ , n } with the property that the resulting edge labels are also distinct, where an edge incident with vertices u and v is assigned the label | f ( u ) − f ( v ) | . A graph which admits a graceful labeling is called a graceful graph. In this paper, inspired by Koh [K.M. Koh, D.G. Rogers and T. Tan, Two theorems on graceful trees, Discrete Math., 25 (1979), 141–148] method, which combines a known graceful trees to obtain a larger graceful trees, we introduced a new method of combining graceful trees called recursive attachment method, and we show that the recursively attached tree T i = T i − 1 ⊕ T A i − 1 is graceful, for i ≥ 1 , where T 0 is a base tree which is taken as a caterpillar and T A i − 1 is an attachment tree which taken as any caterpillar. Here T i − 1 ⊕ T A i − 1 represents a tree obtained by attaching a copy of T A i − 1 at each vertex of degree at least two in T i − 1 , for i ≥ 1 . Consequently the graceful tree conjecture is true for every recursively attached caterpillar tree T i , for i ≥ 1 .

Cycle Partition of Two-Connected and Two-Edge Connected Graphs

Let G be any graph. Then c(G) denotes the circumference of G and is defined as follows: if G is edgeless then c(G) = 1; G is acyclic but G contains at least one edge, then c(G) = 2, if G contains a cycle then c(G) denotes the length of a longest cycle in G. A graph G is called (c1, c2)-partitionable for a pair of positive integers (c1, c2), if c1 + c2 = c(G) and the vertex set V(G) admits a partition (V1, V2) such that