In this article we present theoretical and computational results on the existence of polyhedral embeddings graphs. The emphasis is cubic We also describe an efficient algorithm to compute all a given graph constructions for graphs with some special properties their embeddings. Some key are that even embedding torus can have in arbitrarily high genus, fact genus {\em close} maximum number vertic...

Let G be a connected simple cubic graph; Aut G denotes its automorphism group. Let n be half the number of vertices of G. We define the arithmetic genus of a (possibly disconnected) graph as e−v+1 where e is the number of edges, v the number of vertices of G). For a connected simple cubic graph, g = n+1. The definition of arithmetic genus is motivated by the following: to a projective nodal cur...

Let v(G) and γ(G) denote the number of vertices and the domination number of a graph G, respectively, and let ρ(G) = γ(G)/v(G). In 1996 B. Reed conjectured that ifG is a cubic graph, then γ(G) ≤ ⌈v(G)/3⌉. In 2005 A. Kostochka and B. Stodolsky disproved this conjecture for cubic graphs of connectivity one and maintained that the conjecture may still be true for cubic 2-connected graphs. Their mi...

Let H be a cubic graph admitting a 3-edge-coloring c : E(H) → Z3 such that the edges colored with 0 and μ ∈ {1, 2} induce a Hamilton circuit of H and the edges colored with 1 and 2 induce a 2-factor F . The graph H is semi-Kotzig if switching colors of edges in any even subgraph of F yields a new 3-edge-coloring ofH having the same property as c . A spanning subgraphH of a cubic graph G is call...

A (1, 2)-eulerian weight w of a cubic graph is hamiltonian if every faithful circuit cover of the graph with respect to w is a set of two Hamilton circuits. Let G be a 3-connected cubic graph containing no Petersen-minor. It is proved in this paper that G admits a Hamilton weight if and only if G can be obtained from K4 by a series of 4 ↔ Y -operations. As a byproduct of the proof of the main t...

An oriented k-coloring of an oriented graph ~ G = (V, ~ E) is a partition of V into k subsets such that there are no two adjacent vertices belonging to the same subset and all the arcs between a pair of subsets have the same orientation. The decision problem k-oriented chromatic number (ocnk) consists of an oriented graph ~ G and an integer k > 0, plus the question if there exists an oriented k...

We describe a generative model for graph edges under specific degree distributions which admits an exact and efficient inference method for recovering the most likely structure. This binary graph structure is obtained by reformulating the inference problem as a generalization of the polynomial time combinatorial optimization problem known as b-matching, which recovers a degree constrained maxim...

Let G be a cubic graph and the graph 2G is obtained by replacing each edge of G with a pair of parallel edges. A proper 6-edgecoloring of 2G is called a Fulkerson coloring of G. It was conjectured by Fulkerson that every bridgeless cubic graph has a Fulkerson coloring. In this paper we show that for a Petersen-minor free Graph G, G is uniquely Fulkerson colorable if and only if G constructed fr...

We show that the 56-vertex Klein cubic graph Γ′ can be obtained from the 28-vertex Coxeter cubic graph Γ by ’zipping’ adequately the squares of the 24 7-cycles of Γ endowed with an orientation obtained by considering Γ as a C-ultrahomogeneous digraph, where C is the collection formed by both the oriented 7-cycles ~ C7 and the 2-arcs ~ P3 that tightly fasten those ~ C7 in Γ. In the process, it i...