In this paper, we prove that the square graph of bistar Bn,n, the splitting graph of Bn,n and the splitting graph of star K1,n are graceful graphs. We also prove that the splitting graph and the shadow graph of bistar Bn,n admit odd graceful labeling.

We consider the class A of graphs that contain no odd hole, no antihole of length at least 5, and no “prism” (a graph consisting of two disjoint triangles with three disjoint paths between them) and the class A′ of graphs that contain no odd hole, no antihole of length at least 5, and no odd prism (prism whose three paths are odd). These two classes were introduced by Everett and Reed and are r...

Abstract Let $M$ be a closed, odd GKM$_3$ manifold of non-negative sectional curvature. We show that in this situation one can associate an ordinary abstract graph to and prove if is orientable, then both the equivariant rational cohomology split off odd-dimensional sphere.

Several NP-hard optimization problems, like Maximum Independent Set, q-Coloring, and Max-Cut are polynomial time solvable on bipartite graphs. An equivalent characterization of bipartite graphs is that it is the set of all graphs that do not contain any odd length cycle. Thus, a natural question here is what happens to the complexity of these problems if we know that the length of the longest o...

The NP-complete problem Feedback Vertex Set is that of deciding whether or not it is possible, for a given integer k ≥ 0, to delete at most k vertices from a given graph so that what remains is a forest. The variant in which the deleted vertices must form an independent set is called Independent Feedback Vertex Set and is also NP-complete. In fact, even deciding if an independent feedback verte...

We prove that any r-coloring of the edges of Km contains a monochromatic even cycle, where m = 3r + 1 if r is odd and m = 3r if r is even. We also prove that Km−1 has an r-coloring without monochromatic even cycles. An easy exercise, perhaps folkloristic, says that in any r-coloring of the edges of K2r+1 there is a monochromatic odd cycle (and this is not true for K2r). This note explores what ...

Let G be a graph of su¢ ciently large order n; and let the largest eigenvalue (G) of its adjacency matrix satis…es (G) > p bn2=4c: Then G contains a cycle of length t for every t n=320: This condition is sharp: the complete bipartite graph T2 (n) with parts of size bn=2c and dn=2e contains no odd cycles and its largest eigenvalue is equal to p bn2=4c: This condition is stable: if (G) is close t...

Call a function f : F 2 → {0, 1} odd-cycle-free if there are no x1, . . . , xk ∈ Fn2 with k an odd integer such that f(x1) = · · · = f(xk) = 1 and x1 + · · · + xk = 0. We show that one can distinguish odd-cycle-free functions from those ǫ-far from being odd-cycle-free by making poly(1/ǫ) queries to an evaluation oracle. To obtain this result, we use connections between basic Fourier analysis an...

We consider two combinatorially simple alternatives to summation with even-degree edge sets for characterizing the class of edge sets E in Kn that have specified odd vertices: replacing a path inside E by a path outside it, or summing with a circle contained in E or in Ec. The latter is equivalent to summation with even-degree edge sets for almost all n, and the former is not quite similarly eq...

Assume that n and k are positive integers with n ≥ 2k + 1. A non-hamiltonian graph G is hypo hamiltonian if G − v is hamiltonian for any v ∈ V (G). It is proved that the generalized Petersen graph P (n, k) is hypo hamiltonian if and only if k = 2 and n ≡ 5 (mod 6). Similarly, a hamiltonian graph G is hyper hamiltonian if G−v is hamiltonian for any v ∈ V (G). In this paper, we will give some nec...