Euler Genus Research Articles

2‐connected 7‐coverings of 3‐connected graphs on surfaces

AbstractAn m‐covering of a graph G is a spanning subgraph of G with maximum degree at most m. In this paper, we shall show that every 3‐connected graph on a surface with Euler genus k ≥ 2 with sufficiently large representativity has a 2‐connected 7‐covering with at most 6k − 12 vertices of degree 7. We also construct, for every surface F2 with Euler genus k ≥ 2, a 3‐connected graph G on F2 with arbitrarily large representativity each of whose 2‐connected 7‐coverings contains at least 6k − 12 vertices of degree 7. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 26–36, 2003

Long Cycles in Graphs on a Fixed Surface

We prove that there exists a function a:N0×R+→N such that (i)If G is a 4-connected graph of order n embedded on a surface of Euler genus g such that the face-width of G is at least a(g, ε), then G can be covered by two cycles each of which has length at least (1−ε)n. We apply this to derive lower bounds for the length of a longest cycle in a graph G on any fixed surface. Specifically, there exist functions b:N0→N and c:N0→R+ such that for every graph G on n vertices that is embedded on a surface of Euler genus g the following statements hold: (ii)If G is 4-connected, then G contains a collection of at most b(g) paths which cover all vertices of G, and G contains a cycle of length at least n/b(g). (iii)If G is 3-connected, then G contains a cycle of length at least c(g)nlog2/log3. Moreover, for each ε>0, every 4-connected graph G with sufficiently large face-width contains a spanning tree of maximum degree at most 3 and a 2-connected spanning subgraph of maximum degree at most 4 such that the number of vertices of degree 3 or 4 in either of these subgraphs is at most ε|V(G)|.

Coloring Locally Bipartite Graphs on Surfaces

It is proved that there is a function f:N→N such that the following holds. Let G be a graph embedded in a surface of Euler genus g with all faces of even size and with edge-width ⩾f(g). Then (i)If every contractible 4-cycle of G is facial and there is a face of size >4, then G is 3-colorable. (ii)If G is a quadrangulation, then G is not 3-colorable if and only if there exist disjoint surface separating cycles C1, …, Cg such that, after cutting along C1, …, Cg, we obtain a sphere with g holes and g Möbius strips, an odd number of which is nonbipartite. If embeddings of graphs are represented combinatorially by rotation systems and signatures [5], then the condition in (ii) is satisfied if and only if the geometric dual of G has an odd number of edges with negative signature.

On the critical point‐arboricity graphs

AbstractIn this paper, we study the critical point‐arboricity graphs. We prove two lower bounds for the number of edges of k‐critical point‐arboricity graphs. A theorem of Kronk is extended by proving that the point‐arboricity of a graph G embedded on a surface S with Euler genus g = 2, 5, 6 or g ≥ 10 is at most $k =\lfloor {9 +\sqrt {1+ 24\, g}\over 4}\rfloor$ with equality holding iff G contains either K2k−1 or K2k−4 + C5 as a subgraph. It is also proved that locally planar graphs have point‐arboricity ≤ 3 and that triangle‐free locally planar‐graphs have point‐arboricity ≤ 2. © 2002 John Wiley & Sons, Inc. J Graph Theory 39: 50–61, 2002

Flexibility of Polyhedral Embeddings of Graphs in Surfaces

Whitney's theorem states that 3-connected planar graphs admit essentially unique embeddings in the plane. We generalize this result to embeddings of graphs in arbitrary surfaces by showing that there is a function ξ:N0→N0 such that every 3-connected graph admits at most ξ(g) combinatorially distinct embeddings of face-width ⩾3 into surfaces whose Euler genus is at most g.

The Spectral Radius of Graphs on Surfaces

This paper provides new upper bounds on the spectral radius ρ (largest eigenvalue of the adjacency matrix) of graphs embeddable on a given compact surface. Our method is to bound the maximum rowsum in a polynomial of the adjacency matrix, using simple consequences of Euler's formula. Let γ denote the Euler genus (the number of crosscaps plus twice the number of handles) of a fixed surface Σ. Then (i) for n⩾3, every n-vertex graph embeddable on Σ has ρ⩽2+2n+8γ−6, and (ii) a 4-connected graph with a spherical or 4-representative embedding on Σ has ρ⩽1+2n+2γ−3. Result (i) is not sharp, as Guiduli and Hayes have recently proved that the maximum value of ρ is 3/2+2n+o(1) as n→∞ for graphs embeddable on a fixed surface. However, (i) is the only known bound that is computable, valid for all n⩾3, and asymptotic to 2n like the actual maximum value of ρ. Result (ii) is sharp for the sphere or plane (γ=0), with equality holding if and only if the graph is a “double wheel” 2K1+Cn−2 (+denotes join). For other surfaces we show that (ii) is within O(1/n1/2) of sharpness. We also show that a recent bound on ρ by Hong can be deduced by our method.

Dirac's map-color theorem for choosability

It is proved that the choice number of every graph G embedded on a surface of Euler genus e ≥ 1 and e ≠ 3 is at most the Heawood number $H(\epsilon)= \lfloor(7+\sqrt{24\epsilon+1})/2\rfloor$ and that the equality holds if and only if G contains the complete graph KH(e) as a subgraph. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 327–339, 1999

On the minimal genus of 2-complexes

Gross and Rosen asked if the genus of a 2-dimensional complex K embeddable in some (orientable) surface is equal to the genus of the graph of appropriate barycentric subdivision of K. We answer the nonorientable genus and the Euler genus versions of Gross and Rosen's question in affirmative. We show that this is not the case for the orientable genus by proving that taking ⌊ log2 g⌋ th barycentric subdivision is not sufficient, where g is the genus of K. On the other hand, (1+⌈log2(g+2)⌉)th subdivision is proved to be sufficient. © 1997 John Wiley & Sons, Inc.

The crossing number ofK3,n in a surface

In this article, we show that the crossing number of K3,n in a surface with Euler genus ϵ is ⌊n/(2ϵ + 2)⌋ (n − (ϵ + 1) {1 + ⌊n/(2ϵ + 2)⌋}). This generalizes a result of Guy and Jenkyns, who obtained this result for the torus. © 1996 John Wiley & Sons, Inc.

The crossing number of K3, n in a surface

In this article, we show that the crossing number of K3,n in a surface with Euler genus ϵ is ⌊n/(2ϵ + 2)⌋ (n − (ϵ + 1) {1 + ⌊n/(2ϵ + 2)⌋}). This generalizes a result of Guy and Jenkyns, who obtained this result for the torus. © 1996 John Wiley & Sons, Inc.

Characterization of the maximum genus of a signed graph

A signed graph is a graph in which each edge is labelled with a sign. A cycle C of a signed graph is balanced if the product of signs on C is positive. By an embedding of a signed graph G on a closed surface S we mean a topological embedding of the corresponding unsigned graph satisfying the following additional condition: A cycle C of G is embedded as an orientation-preserving cycle on S if and only if C is balanced. Let ε M( G) be the maximum Euler genus of a closed surface in which the signed graph G has a 2-cell embedding. We establish a minimax formula for ε M( G). The method used provides a unification of several results in maximum genus of unsigned graphs. For example, Xuong's and Nebeský's formulas for the maximum orientable genus of a graph and the Ringel-Stahl singleface nonorientable embedding theorem are corollaries of the work in this paper.

On the Euler genus of a 2-connected graph

The Euler genus of the surface Σ obtained from the sphere by the addition of k crosscaps and h handles is ε( Σ) = k + 2 h. For a graph G, the Euler genus ε( G) of G is the smallest Euler genus among all surfaces in which G embeds. The following additivity theorem is proved. Theorem. Suppose G = H ∪ K, where H and K have exactly the vertices v and w in common. Then ε( G) = min( ε( H + vw) + ε( K + vw), ε( H) + ε( K) + 2).

On the non-orientable genus of a 2-connected graph

In earlier works, additivity theorems for the genus and Euler genus of unions of graphs at two points have been given. In this work, the analogous result for the non-orientable genus is given. If Σ is obtained from the sphere by the addition of k>0 crosscaps, define γ(Σ) to be k. For a graph G, define γ(G) to be the least element in the set {γ(Σ) | G embeds in Σ}. Theorem. Let H 1 and H 2 be connected graphs such that H 1 ∩ H 2 consists of the isolated vertices v and w. Then, for some μ ϵ −1, 0, 1, 2, γ(H 1 ∪ H 2) = γ(H 1) + γ(H 2) + μ. A formula for μ is given.