A graph G with q edges and p vertices is edge-graceful if some bijection f from E(G) onto {1,...,q} makes the induced vertex sums f^+(v), the sum of f(e) over the edges e incident to v, distinct modulo p. Lee conjectured in 1989 that every tree of odd order is edge-graceful; the broadest general result we have located, due in equivalent form to Kaplan, Lev and Roditty, covers trees of odd order with at most one vertex of degree two. We prove that every tree of odd order with at most two vertices of degree two is edge-graceful. The proof combines zero-sum block partitions of Z_n with perfect and hooked Langford sequences; the residual case analysis is verified symbolically for all odd n<= 5001 and holds uniformly beyond, and the construction was executed and independently re-checked on all 2,245,070 trees of odd order at most 25 with exactly two vertices of degree two. Since an edge-graceful graph is antimagic, the theorem also enlarges the family of trees known to be antimagic.
A bisection of a graph is a partition of its vertex set into two parts whose sizes differ by at most one. We prove that every graph with $m$ edges and maximum degree at most $k$ has a bisection of size at least $\frac{k+1}{2k}m$ when $k$ is odd, and at least $\frac{k+2}{2(k+1)}m$ when $k$ is even. Both bounds are sharp...
Let G(V,E) be a graph, and let f: V(G) →{1,2,….|𝑉(𝐺)|} be a bijective vertex labeling. The induced edge labeling f*: E(G) → {0,1} is defined by assigning the label 1 to an edge uv whenever the labels f(u) and f(v) have same parity, and the label 0 whenever they have different parities. If the difference between the num...
S. N. Kumar, D. Bharathi· International Journal of Mat...· 0 citations
For a fixed graph F, the F-degree of a vertex v in a host graph H is the number of subgraphs of H isomorphic to F that contain v, and H is F-irregular if its F-degrees are pairwise distinct. We show that every finite connected graph F on at least three vertices admits a finite connected F-irregular host. For noncomplet...
A graph is \emph{elementary} if its edges can be coloured with two colours so that the two edges of every induced path on three vertices receive different colours. Elementary graphs are one of the two basic classes in the Chv\'atal--Sbihi decomposition of claw-free perfect graphs; they include all line graphs of bipart...
N. Narayanan, Nandana K. Vasudevan, K. Somasundaram· 0 citations
We prove that the realization graph of every graphical degree sequence is maximally Hamiltonian: it is Hamilton-laceable when bipartite on more than one vertex, and Hamilton-connected otherwise. This answers Problem P59 of M\"utze's survey of combinatorial Gray codes, and the Hamiltonicity question recorded as open by...
