Given a graph G, the acyclic orientation graph of G, denoted AO(G), is the graph whose vertices are the acyclic orientations of G, and two acyclic orientations are joined by an edge in AO(G) iff they differ by the reversal of a single edge. A hamilton cycle in AO(G) gives a Gray code listing of the acyclic orientations of G. We prove that for certain graphs G, AO(G) is hamiltonian, and give exp...

In 1954, Tutte conjectured that every bridgeless graph has a nowherezero 5-flow. Let ω be the minimum number of odd cycles in a 2-factor of a bridgeless cubic graph. Tutte’s conjecture is equivalent to its restriction to cubic graphs with ω ≥ 2. We show that if a cubic graph G has no edge cut with fewer than 5 2ω − 1 edges that separates two odd cycles of a minimum 2-factor of G, then G has a n...

A long-standing conjecture asserts the existence of a positive constant c such that every simple graph of order n without isolated vertices contains an induced subgraph of order at least cn such that all degrees in this induced subgraph are odd. Radcliffe and Scott have proved the conjecture for trees, essentially with the constant c = 2/3. Scott proved a bound for c depending on the chromatic ...

An odd hole is an induced odd cycle of length at least 5. Scott and Seymour confirmed a conjecture of Gyárfás and proved that if a graph G has no odd holes then χ(G) 6 22 ω(G)+2 . Chudnovsky, Robertson, Seymour and Thomas showed that if G has neither K4 nor odd holes then χ(G) 6 4. In this note, we show that if a graph G has neither triangles nor quadrilaterals, and has no odd holes of length a...

A shell graph is the join of a path P k of 'k' vertices and K 1. A subdivided shell graph can be constructed by subdividing the edges in the path of the shell graph. In this paper we prove that the disjoint union of two subdivided shell graphs is odd graceful and also one modulo three graceful. 1. Introduction A graph labeling is an assignment of integers to the vertices or edges, or both, subj...

A binary clutter is cycling if its packing and covering linear program have integral optimal solutions for all Eulerian edge capacities. We prove that the clutter of odd stwalks of a signed graph is cycling if and only if it does not contain as a minor the clutter of odd circuits of K5 nor the clutter of lines of the Fano matroid. Corollaries of this result include, of many, the characterizatio...

Let G = (V, E) be a finite, simple and undirected graph. A graph G with q edges is said to be odd-graceful if there is an injection f : V (G) → {0, 1, 2, . . . , 2q− 1} such that, when each edge xy is assigned the label |f (x)− f (y)| , the resulting edge labels are {1, 3, 5, . . . , 2q− 1} and f is called an odd graceful labeling of G. Motivated by the work of Z. Gao [6] in which he studied th...

For graphs G and H, a map f : V (G) 7→ V (H) is a homomorphism if f preserves adjacency. Let Hom(G,H) denote the set of all homomorphisms from G into H. In this paper, we proved that for a simple graph G with n = |V (G)| and for k with n ≥ k ≥ 5, if the odd girth of G is at least 2k+1 and if the minimum degree δ(G) > 2n/(2k+3), then Hom(G,Z2k+1) 6= ∅, where Z2k+1 denotes the cycle of length 2k ...