Maximum Genus Research Articles

Maximum genus and chromatic number of graphs

Let T be a spanning tree of a connected graph G. Denote by ξ( G, T) the number of components in G⧹ E( T) with odd number of edges. The value min T ξ(G,T) is known as the Betti deficiency of G, denoted by ξ( G), where the minimum is taken over all spanning trees T of G. It is known (N.H. Xuong, J. Combin. Theory 26 (1979) 217–225) that the maximum genus of a graph is mainly determined by its Betti deficiency ξ( G). Let G be a k-edge-connected graph ( k⩽3) whose complementary graph has the chromatic number m. In this paper we prove that the Betti deficiency ξ( G) is bounded by a function f k ( m) on m, and the bound is the best possible. Thus by Xuong's maximum genus theorem we obtain some new results on the lower bounds of the maximum genus of graphs.

Maximum genus of a graph in terms of its embedding properties

Let G be a graph that is cellularly embedded in the projective plane such that the dual graph is Hamiltonian. Then we prove that G is upper embeddable. In the meantime we also obtain the same result for other general orientable surfaces if the dual graph contains a separating Hamilton circuit.

The Maximum Genus of a Graph with Given Diameter and Connectivity

The Maximum Genus of a Graph with Given Diameter and Connectivity

The Maximum Genus of a Graph with Given Diameter and Connectivity

Abstract In this paper, we first review some of the known results about the maximum genus of a graph with given diameter or (and) connectivity. Then we prove that a 3-connected diameter 4 multigraph has Betti deficiency at most 2. Furthermore, we show this upper bound is sharp.

Embedding Digraphs on Orientable Surfaces

We consider a notion of embedding digraphs on orientable surfaces, applicable to digraphs in which the indegree equals the outdegree for every vertex, i.e., Eulerian digraphs. This idea has been considered before in the context of compatible Euler tours or orthogonal A-trails by Andersen and by Bouchet. This prior work has mostly been limited to embeddings of Eulerian digraphs on predetermined surfaces and to digraphs with underlying graphs of maximum degree at most 4. In this paper, a foundation is laid for the study of all Eulerian digraph embeddings. Results are proved which are analogous to those fundamental to the theory of undirected graph embeddings, such as Duke's theorem [5], and an infinite family of digraphs which demonstrates that the genus range for an embeddable digraph can be any nonnegative integer given. We show that it is possible to have genus range equal to one, with arbitrarily large minimum genus, unlike in the undirected case. The difference between the minimum genera of a digraph and its underlying graph is considered, as is the difference between the maximum genera. We say that a digraph is upper-embeddable if it can be embedded with two or three regions and prove that every regular tournament is upper-embeddable.

A tight lower bound on the maximum genus of a 3-connected loopless multigraph

It is proved that every 3-connected loopless multigraph has maximum genus at least one-third of its cycle rank plus one if its cycle rank is not less than ten, and if its cycle rank is less than ten, it is upper-embeddable. This lower bound is tight. There are infinitely many 3-connected loopless multigraphs attaining this bound.

Lower bounds on the maximum genus of loopless multigraphs

The lower bounds on the maximum genus of loopless graphs are obtained according to the connectivity of these graphs. This not only answers a question of Chen, Archdeacon and Gross, but also generalizes the previous known results. Thus, a picture of the lower bounds on the maximum genus of loopless multigraphs is presented.

The decay number and the maximum genus of diameter 2 graphs

Let ζ(G) (resp. ξ(G)) be the minimum number of components (resp. odd size components) of a co-tree of a connected graph G. For every 2-connected graph G of diameter 2, it is known that m(G)⩾2n(G)−5 and ξ(G)⩽ζ(G)⩽4. These results define three classes of extremal graphs. In this paper, we prove that they are the same, with the exception of loops added to vertices.

Face Size and the Maximum Genus of a Graph 1. Simple Graphs

This paper shows that a simple graph which can be cellularly embedded on some closed surface in such a way that the size of each face does not exceed 7 is upper embeddable. This settles one of two conjectures posed by Nedela and Škoviera (1990, in “Topics in Combinatorics and Graph Theory,” pp. 519–529, Physica Verlag, Heidelberg). The other conjecture will be proved in a sequel to this paper.

A relative maximum genus graph embedding and its local maximum genus

A relative embedding of a connected graph is an embedding of the graph in some surface with respect to some closed walks, each of which bounds a face of the embedding. The relative maximum genus of a connected graph is the maximum of integerk with the property that the graph has a relative embedding in the orientable surface withk handles. A polynomial algorithm is provided for constructing relative maximum genus embedding of a graph if the relative tree of the graph is planar. Under this condition, just like maximum genus embedding, a graph does not have any locally strict maximum genus.

A note on the maximum genus of 3-edge-connected nonsimple graphs

Let G be a 3-edge-connected graph (possibly with multiple edges or loops), and let γM(G) and β(G) be the maximum genus and the Betti number of G, respectively. Then γM(G) can be proved and this answers a question posed by Chen, et al. in 1996.

Maximum Genus, Girth and Connectivity

In this paper, the lower bounds of maximum genera of simplicial graphs under the constraints of girth and connectivity are discussed extensively. A complete picture on the lower bounds of maximum genera of graphs is formed. There are infinitely many graphs with the maximum genera attaining these lower bounds.

The Maximum Genus on a 3-Vertex-Connected Graph

This paper shows that the lower bound on the maximum genus for a 3-vertex-connected graph G, which may have multiple edges and loops, is at least ⅓β(G). This answers the question posed by the authors in [9].

On the Lower Bounds of the Maximum Genus of Graphs

On the Lower Bounds of the Maximum Genus of Graphs

Maximum Genus, Independence Number and Girth

It is known (for example see [2]) that the maximum genus of a graph is mainly determined by the Betti deficiency of the graph. In this paper, the authors establish an upper bound on the Betti deficiency in terms of the independence number as well as the girth of a graph, and thus use the formulation in [2] to translate this result to lower bound on the maximum genus. Meantime it is shown that both of the bounds are best possible.

A tight lower bound on the maximum genus of 3-edge connected loopless graphs

It is shown that the lower bound on the maximum genus of a 3-edge connected loopless graph is at least one-third of its cycle rank. Moreover, this lower bound is tight. There are infinitely many such graphs attaining the bound.

The maximum genus of a 3-regular simplicial graph

In this paper, the relationship between non-separating independent number and the maximum genus of a 3-regular simplicial graph is presented. A lower bound on the maximum genus of a 3-regular graph involving girth is provided. The lower bound is tight, it improves a bound of Huang and Liu.

The maximum genus of graphs with diameter three

This paper shows that if G is a simple graph with diameter three then G is up-embeddable unless G is either a Δ 2-graph (Fig. 1) or a Δ 3-graph (Fig. 2) with ξ( G) = 2, i.e., the maximum genus γ M ( G) = ( β( G) − 2)/2.

Maximum genus and girth of graphs

In this paper, a lower bound on the maximum genus of a graph in terms of its girth is established as follows: let G be a simple graph with minimum degree at least three, and let g be the girth of G. Then γ(G)⩾ g−2 2(g−1) β(G)+ 1 g−1 except for G=K 4, where β( G) denotes the cycle rank of G and K 4 is the complete graph with four vertices.

The maximum genus, matchings and the cycle space of a graph

In this paper we determine the maximum genus of a graph by using the matching number of the intersection graph of a basis of its cycle space. Our result is a common generalization of a theorem of Glukhov [5] and a theorem of Nebeský [15].