Genus of a graph and its strong preservers

Mathematics

A graph has genus k if it can be embedded without edge crossings on a smooth orientable surface of genus k and not on one of genus k − 1 . A mapping of the set of graphs on n vertices to itself is called a linear operator if the image of a union of graphs is the union of their images and if it maps the edgeless graph to the edgeless graph. We investigate linear operators on the set of graphs on n vertices that map graphs of genus k to graphs of genus k and graphs of genus k + 1 to graphs of genus k + 1 . We show that such linear operators are necessarily vertex permutations. Similar results with different restrictions on the genus k preserving operators give the same conclusion.

Mathematics

If a graph can be embedded in a smooth orientable surface of genus g without edge crossings and can not be embedded on one of genus g − 1 without edge crossings, then we say that the graph has genus g. We consider a mapping on the set of graphs with m vertices into itself. The mapping is called a linear operator if it preserves a union of graphs and it also preserves the empty graph. On the set of graphs with m vertices, we consider and investigate those linear operators which map graphs of genus g to graphs of genus g and graphs of genus g + j to graphs of genus g + j for j ≤ g and m sufficiently large. We show that such linear operators are necessarily vertex permutations.

Journal of Combinatorial Theory, Series B, 1985

This paper classifies the regular imbeddings of the complete graphs K,, in orientable surfaces. Biggs showed that these exist if and only if n is a prime power p', his examples being Cayley maps based on the finite field F= GF(n). We show that these are the only examples, and that there are q5(n-1)/e isomorphism classes of such maps (where 4 is Euler's function), each corresponding to a conjugacy class of primitive elements of F, or equivalently to an irreducible factor of the cyclotomic polynomial Qn-r(z) over GF(p). We show that these maps are all equivalent under Wilson's map-operations Hi, and we determined for which n they are reflexible or self-dual.

Journal of Combinatorial Theory, 2007

We show that for n=4 and n⩾6, Kn has a nonorientable embedding in which all the facial walks are hamilton cycles. Moreover, when n is odd there is such an embedding that is 2-face-colorable. Using these results we consider the join of an edgeless graph with a complete graph, Km¯+Kn=Km+n−Km, and show that for n⩾3 and m⩾n−1 its nonorientable genus

Ars Combinatoria Waterloo Then Winnipeg, 1997

The star graph S n is a graph with S n the set of all permutations over f1; : : :; ng as its vertex set; two vertices 1 and 2 are connected if 1 can be obtained form 2 by swapping the rst element of 1 with one of the other n ? 1 elements. In this paper we establish the genus of the star graph. We show that the genus, g n of S n , is exactly equal to n!(n?4)=6+1 by establishing a lower bound and inductively giving a drawing on a surface of appropriate genus.

Journal of Combinatorial Theory, Series B, 2008

A graph G with at least 2n + 2 vertices is said to be n-extendable if every matching of size n in G extends to a perfect matching. It is shown that (1) if a graph is embedded on a surface of Euler characteristic χ , and the number of vertices in G is large enough, the graph is not 4-extendable; (2) given g > 0, there are infinitely many graphs of orientable genus g which are 3-extendable, and given g 2, there are infinitely many graphs of non-orientable genus g which are 3-extendable; and (3) if G is a 5-connected triangulation with an even number of vertices which has genus g > 0 and sufficiently large representativity, then it is 2-extendable.

Discrete Mathematics, 1998

A conjecture of Robertson and Thomas on the orientable genus of graphs with a given nonorientable embedding is disproved.

Journal of Combinatorial Theory, Series B, 2006

In 1976, Stahl and White conjectured that the nonorientable genus of Kl,m,n, where l ≥ m ≥ n, is (l−2)(m+n−2) 2 ¡ . The authors recently showed that the graphs K3,3,3 , K4,4,1, and K4,4,3 are counterexamples to this conjecture. Here we prove that apart from these three exceptions, the conjecture is true. In the course of the paper we introduce a construction called a transition graph, which is closely related to voltage graphs.

Discrete Mathematics, 1989

It is shown that the genus of an embedding of a graph can be determined by the rank of a certain matrix. Several applications to problems involving the genus of graphs are presented.

Pacific Journal of Mathematics, 1972

Given two graphs G and H, a new graph G(H), called the composition (or lexicographic product) of G and H, can be formed. In this paper, a formula is developed to give the genus for a large class of lexicographic products. In the simplest special case, the genus of the product is given by the first Betti number of one of the factors.