The main contributions of this paper are three-fold. First, we use a dynamic approach based on Reiter’s pioneering work on Karp-Miller computation graphs [19] to give a new and short proof of Mohar’s Minty-type Theorem [15]. Second, we bridge circular colorings and discrete event dynamic systems to show that the Barbosa and Gafni’s results on circular chromatic number [5, 21] can be generalized...

Menger's Theorem states that in a 3-connected graph, any two vertices are joined by three openly disjoint paths. Here we consider 3-connected cubic graphs where two vertices exist so that the three disjoint paths between them contain all of the vertices of the graph (we call these graphs 3*connected); and also where the latter is true for ALL pairs of vertices (globally 3*-connected). A necessa...

The generalized Petersen graph GP(n, k), n > 2 and 1 < k & n 1, has vertex-set (uO,ul ,..., u *-,, vO,u ,,..., v,_,) and edge-set (u,ui+,,uioi,vivi+,:O 8.

Let G = (V, E) be a graph. A function f : V → {−1,+1} defined on the vertices of G is a signed total dominating function if the sum of its function values over any open neighborhood is at least one. The signed total domination number of G, γ t (G), is the minimum weight of a signed total dominating function of G. In this paper, we study the signed total domination number of generalized Petersen...

Partial cubes are graphs isometrically embeddable into hypercubes. In this paper it is proved that every cubic, vertex-transitive partial cube is isomorphic to one of the following graphs: K2ƒC2n, for some n ≥ 2, generalized Petersen graph G(10, 3), permutahedron, truncated cuboctahedron, or truncated icosidodecahedron. This complete classification of cubic, vertex-transitive partial cubes, a f...

The matching preclusion number of a graph G, mp(G), is the minimum number of edges whose deletion leaves a resulting graph that has neither perfect matchings nor almost perfect matchings. Besides its theoretical linkage with conditional connectivity and extremal graph theory, the matching preclusion number is a measure of robustness in interconnection networks. In this paper we develop general ...

A double Roman dominating function on a graph G=(V,E) is f:V?{0,1,2,3}, satisfying the condition that every vertex u for which f(u)=1 adjacent to at least one assigned 2 or 3, and with f(u)=0 3 two vertices 2. The weight of f equals sum w(f)=?v?Vf(v). minimum G called domination number ?dR(G) G. We obtain tight bounds in some cases closed expressions generalized Petersen graphs P(ck,k). In shor...