An injective map f : E(G) → {±1, ±2, · · · , ±q} is said to be an edge pair sum labeling of a graph G(p, q) if the induced vertex function f*: V (G) → Z − {0} defined by f*(v) = (Sigma e∈Ev) f (e) is one-one, where Ev denotes the set of edges in G that are incident with a vetex 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} accordin...

The identification of diametrical vertices in the d-dimensional hypercube (d >/3) leads to a (0, 2)-graph of degree d on 2 d1 vertices and of diameter [. d/2 J namely the extended odd graph (or Laborde-Mulder graph) for odd values of d, and the half-cube for even values of d. In this paper we prove that the diameter of a (0, 2)-graph of degree d on 2 ~~ vertices is at least [. d/2 .~ and when d...

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} defined 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 as p is even or o...

It is an old problem in graph theory to test whether a graph contains a chordless cycle of length greater than three (hole) with a specific parity (even, odd). Studying the structure of graphs without odd holes has obvious implications for Berge’s strong perfect graph conjecture that states that a graph G is perfect if and only if neither G nor its complement contain an odd hole. Markossian, Ga...

A snark is a cubic cyclically 4–edge connected graph with edge chromatic number four and girth at least five. We say that a graph G is odd 2–factored if for each 2–factor F of G each cycle of F is odd. In this paper, we present a method for constructing odd 2–factored snarks. In particular, we construct two new odd 2–factored snarks that disprove a conjecture by some of the authors. Moreover, w...

Normal graphs are defined in terms of cross-intersecting set families: a graph is normal if it admits a clique cover Q and a stable set cover S s.t. every clique in Q intersects every stable set in S. Normal graphs can be considered as closure of perfect graphs by means of co-normal products (K ̈orner [6]) and graph entropy (Czisz ́ar et al. [5]). Perfect graphs have been recently characterized a...

Let G be a simple graph and f : V (G) 7→ {1, 3, 5, ...} an odd integer valued function defined on V (G). A spanning subgraph F of G is called a (1, f)odd factor if dF (v) ∈ {1, 3, ..., f(v)} for all v ∈ V (G), where dF (v) is the degree of v in F . For an odd integer k, if f(v) = k for all v, then a (1, f)odd factor is called a [1, k]-odd factor. In this paper, the structure and properties of a...

The odd edge connectivity of a graph G, denoted by o(G), is the size of a smallest odd edge cut of the graph. Let S be any given surface and be a positive real number. We proved that there is a function fS( ) (depends on the surface S and lim !0 fS( )1⁄41) such that any graph G embedded in S with the odd-edge connectivity at least fS( ) admits a nowhere-zero circular (2þ )-flow. Another major r...