An odd hole in a graph is an induced subgraph which is a cycle of odd length at least five. In 1985, A. Gyárfás made the conjecture that for all t there exists n such that every graph with no Kt subgraph and no odd hole is n-colourable. We prove this conjecture.

In this paper we study oriented bipartite graphs. In particular, several characterizations of bitournaments are obtained. We introduce the concept of odd-even graphs and show that any (oriented) bipartite graph can be represented by some (oriented) odd-even graph. We show that the famous Goldbach’s conjecture is equivalent to the connectedness of certain odd-even graphs.

let g be a (p,q) graph. an injective map f : e(g) → {±1,±2,...,±q} is said to be an edge pair sum labeling if the induced vertex function f*: v (g) → z - {0} defined by f*(v) = σp∈ev f (e) is one-one where ev denotes the set of edges in g that are incident with a vertex v and f*(v (g)) is either of the form {±k1,±k2,...,±kp/2} or {±k1,±k2,...,±k(p-1)/2} u {±k(p+1)/2} according a...

An odd hole in a graph is an induced subgraph which is a cycle of odd length at least five. In 1985, A. Gyárfás made the conjecture that for all t there exists n such that every graph with no Kt subgraph and no odd hole is n-colourable. We prove this conjecture.

In 1991, Gnanajothi [4] proved that the path graph n P with n vertex and 1 n − edge is odd graceful, and the cycle graph m C with m vertex and m edges is odd graceful if and only if m even, she proved the cycle graph is not graceful if m odd. In this paper, firstly, we studied the graph m n C P ∪ when 4, 6,8,10 m = and then we proved that the graph m n C P ∪ is odd graceful if m is even. Finall...

The Skolem machine is a Turing-complete machine model where the instructions are first-order formulas of a specific form. We introduce Skolem machines and prove their logical correctness and completeness. Skolem machines compute queries for the Geolog language, a rich fragment of first-order logic. The concepts of Geolog trees and complete Geolog trees are defined, and these tree concepts are u...

a graph g is said to have a totally magic cordial labeling with constant c if there exists a mapping f : v (g) ∪ e(g) → {0, 1} such that f(a) + f(b) + f(ab) ≡ c (mod 2) for all ab ∈ e(g) and |nf (0) − nf (1)| ≤ 1, where nf (i) (i = 0, 1) is the sum of the number of vertices and edges with label i. in this paper, we give a necessary condition for an odd graph to be not totally magic cordial and ...

Suppose G is a connected, k-regular graph such that Spec…G† ˆ Spec…G† where G is a distance-regular graph of diameter d with parameters a1 ˆ a2 ˆ ˆ adÿ1 ˆ 0 and ad > 0; i.e., a generalized odd graph, we show that G must be distance-regular with the same intersection array as that of G in terms of the notion of Ho ̈man Polynomials. Furthermore, G is isomorphic to G if G is one of the odd polygon ...

The ladder graph plays an important role in many applications as Electronics, Electrical and Wireless communication areas. The aim of this work is to present a new class of odd graceful labeling for the ladder graph. In particular, we show that the ladder graph Ln with m-pendant Ln mk1 is odd graceful. We also show that the subdivision of ladder graph Ln with m-pendant S(Ln)  mk1 is odd grace...

Let G denote a finite simple graph with edge ideal I(G). Letting I(G) denote the Alexander dual of I(G), we show that a description of the induced cycles of G of odd length is encoded in the associated primes of (I(G)). This result forms the basis for an algorithm to detect all the odd induced cycles of a graph via ideal operations, e.g., intersections, products, and colon operations. Moreover,...