Abstract: It is well known that any graph imbedded in the torus has chromatic number at most seven, and that seven is attained by the graph K7. In this note we show that any toroidal graph containing no triangles has chromatic number at most four, and produce an example attaining this upper bound. The results are then extended for arbitrary girth.

Abstract: A graph of genus 2 which is irreducible with respect to this property must have at least eight vertices. It is shown here that there are exactly three such 2-irreducible graphs having eight vertices. The description of these three graphs, together with the Kuratowski theorem, leeds to a determination of the genus of each graph having fewer than nine vertices.

Abstract: Publisher Summary This chapter discusses the non-Hamiltonian planner maps. A planar map is a dissection of the sphere or closed plane into a finite number of simply connected polygonal regions called faces or countries by means of a graph drawn in the surface. It is assumed that this graph has no loop or isthmus. A Hamiltonian circuit in a map is a circuit in its graph passing through every vertex. A map is called Hamiltonian or non-Hamiltonian according to as it does or does not have such a circuit. A Hamiltonian bond in a graph G is a set H of edges such that the rest of the graph consists of two disjoint trees, and each edge of H has one end in each tree. Denoting the number of vertices of G of valency i by f i , and suppose f i ' of these to be in the first tree and f i ’ in the second. This form of the theory applies to all graphs, whether planar or nonplanar.

Abstract: We have discussed planar graphs, their characterization, testing and generation of codes. We have shown that the codes solve the isomorphism and automorphism group problems. Since matroids give a complete theory of duality for graphs, it is to be expected that they yield insights and characterizations for planar graphs. Conversely, a fertile field of research is to generalize known conditions on planar graphs to realizability conditions on matroids. Tutte has provided one such theorem and Welsh another. The wheel algorithm for testing planarity that we discussed yields a third characterization of matroids.

Abstract: A sufficient condition is given for a planar graph to be 4-colorable. This condition is in terms of the sums of the degrees of a subset of the vertex set of the graph.

Abstract: Publisher Summary This chapter discusses the evolution of path number of a graph in context of covering and packing in graphs. The concepts of packing and covering were explored in a lecture given in New York city, as a generalization of path number, arboricity, and several other graphical invariants. This approach suggested the definition of the linear arboricity of a graph, which has an interpretation in file structures. An alternative path-covering invariant of a graph can be defined as the minimum number of paths, unrestricted in that they are not necessarily line disjoint, needed to cover the lines of G. Although, several theorems are there for determining the path number of a graph, still there are some unsolved problems related to it and there are still no effective and convenient computer algorithms for determining the values of these five invariants for a given graph.

Abstract: An algorithm is presented for determining whether or not two planar graphs are isomorphic. The algorithm requires O(V log V) time, if V is the number of vertices in each graph.

Abstract: A conjectured property of bridgeless cubic planar graphs is shown to be equivalent to the four colour conjecture. In establishing this equivalence use is made of the Kbnig.Hall theorem on the existence of one-factors in bipartite graphs. 1. The four colour conjecture. For a discussion of the four colour conjecture and related topics we refer the reader to Ore [3]. We merely content ourselves here with a statement of the conjecture and of two equivalent conjectures (proofs of their equivalence are given in [3]). Our graph-theoretic terminology is that of Harary [2] (although we use vertices and edges for what are respectively called points and lines in [2]). FouR COLOUR CONJECTURE. The faces of any plane graph can be coloured with four colours so that no two incident faces are assigned the same colour. THE HEAWOOD CONJECTURE. For any bridgeless cubic plane graph G there is a mappingf: V(G)-3{1, + 1} so that, for every face F, J,Ff (x)0 (mod 3). THE TAIT CONJECTURE. Every bridgeless cubic planar graph has an even two-factor (that is a two-factor in which each cycle is of even length). 2. Balanced colourings. Let G be a graph with vertex set V(G) and edge set E(G). A partition W= (B, W) of V(G) is a two-colouring of V(G); ' is an equitable two-colouring if IBI = I WI. For a two-colouring ' and a vertex x, we define w(x, '), the weight of x in ', by w(x,jW=-2 if xeB, = +2 if xeW, and for Sc V(G) we denote 2ies w(x, ') by w(S, s). Again, for SC V(G), let v(S) denote the number of edges of G having exactly one end in S. A balanced colouring of G is a two-colouring ' of Received by the editors August 2, 1971. AMS 1970 subject classifications. Primary 05C15.