Coloring Eulerian Triangulations of the Klein Bottle

We show that an Eulerian triangulation of the Klein bottle has chromatic number equal to six if and only if it contains a complete graph of order six, and it is 5-colorable, otherwise. As a consequence of our proof, we derive that every Eulerian triangulation of the Klein bottle with face-width at least four is 5-colorable.     

Coloring even-faced graphs in the torus and the Klein bottle

We prove that a triangle-free graph drawn in the torus with all faces bounded by even walks is 3-colorable if and only if it has no subgraph isomorphic to the Cayley graph C(Z ₁₃; 1,5). We also prove that a non-bipartite quadrangulation of the Klein bottle is 3-colorable if and only if it has no non-contractible separating cycle of length at most four and no odd walk homotopic to a non-contractible two-sided simple closed curve. These results settle a conjecture of Thomassen and two conjectures of Archdeacon, Hutchinson, Nakamoto, Negami and Ota. Institute for Theoretical Computer Science is supported as project 1M0545 by the Ministry of Education of the Czech Republic. The author was visiting Georgia Institute of Technology as a Fulbright scholar in the academic year 2005/06. Partially supported by NSF Grants No. DMS-0200595 and DMS-0354742.     

Spanning eulerian subgraphs of bounded degree in triangulations

We show that every triangulation of a disk or an annulus has a spanning Eulerian subgraph with maximum degree eight. Since every triangulation in the projective plane, the torus and the Klein bottle has a spanning subgraph which triangulates an annulus, this implies that all triangulations in the projective plane, the torus and the Klein bottle have spanning Eulerian subgraphs with maximum degree at most eight.     

Coloring graphs with crossings

We generalize the Five-Color Theorem by showing that it extends to graphs with two crossings. Furthermore, we show that if a graph has three crossings, but does not contain K₆ as a subgraph, then it is also 5-colorable. We also consider the question of whether the result can be extended to graphs with more crossings.     

Edge Partition of Graphs Embeddable in the Projective Plane and the Klein Bottle

In a previous paper by the author joint with Baogang XU published in Discrete Math in 2018, we show that every non-planar toroidal graph can be edge partitioned into a planar graph and an outerplanar graph. This edge partition then implies some results in thickness and outerthickness of toroidal graphs. In particular, if each planar graph has outerthickness at most $2$ (conjectured by Chartrand, Geller and Hedetniemi in 1971 and the confirmation of the conjecture was announced by Gon\c{c}alves in 2005), then the outerthickness of toroidal graphs is at most 3 which is the best possible due to $K_7$. In this paper we continue to study the edge partition for projective planar graphs and Klein bottle embeddable graphs. We show that (1) every non-planar but projective planar graph can be edge partitioned into a planar graph and a union of caterpillar trees; and (2) every non-planar Klein bottle embeddable graph can be edge partitioned into a planar graph and a subgraph of two vertex amalgamation of a caterpillar tree with a cycle with pendant edges. As consequences, the thinkness of projective planar graphs and Klein bottle embeddabe graphs are at most $2$, which are the best possible, and the outerthickness of these graphs are at most $3$.     

Degree-regular triangulations of torus and Klein bottle

A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In [8], Lutz has classified all the weakly regular triangulations on at most 15 vertices. In [5], Datta and Nilakantan have classified all the degree-regular triangulations of closed surfaces on at most 11 vertices. In this article, we have proved that any degree-regular triangulation of the torus is weakly regular. We have shown that there exists ann-vertex degree-regular triangulation of the Klein bottle if and only if n is a composite number ≥ 9. We have constructed two distinctn-vertex weakly regular triangulations of the torus for eachn ≥ 12 and a (4m + 2)-vertex weakly regular triangulation of the Klein bottle for eachm ≥ 2. For 12 ≤n ≤ 15, we have classified all then-vertex degree-regular triangulations of the torus and the Klein bottle. There are exactly 19 such triangulations, 12 of which are triangulations of the torus and remaining 7 are triangulations of the Klein bottle. Among the last 7, only one is weakly regular.     

2-connected coverings of bounded degree in 3-connected graphs

In a recent paper, Barnette showed that every 3-connected planar graph has a 2-connected spanning subgraph of maximum degree at most fifteen, he also constructed a planar triangulation that does not have 2-connected spanning subgraphs of maximum degree five. In this paper, we show that every 3-connected graph which is embeddable in the sphere, the projective plane, the torus or the Klein bottle has a 2-connected spanning subgraph of maximum degree at most six. © 1995 John Wiley & Sons, Inc.     

Chromatic numbers of quadrangulations on closed surfaces

It has been shown that every quadrangulation on any nonspherical orientable closed surface with a sufficiently large representativity has chromatic number at most 3. In this paper, we show that a quadrangulation G on a nonorientable closed surface N_k has chromatic number at least 4 if G has a cycle of odd length which cuts open N_k into an orientable surface. Moreover, we characterize the quadrangulations on the torus and the Klein bottle with chromatic number exactly 3. By our characterization, we prove that every quadrangulation on the torus with representativity at least 9 has chromatic number at most 3, and that a quadrangulation on the Klein bottle with representativity at least 7 has chromatic number at most 3 if a cycle cutting open the Klein bottle into an annulus has even length. As an application of our theory, we prove that every nonorientable closed surface N_k admits an eulerian triangulation with chromatic number at least 5 which has arbitrarily large representativity. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 100–114, 2001     

5-Chromatic even triangulations on surfaces

A triangulation is said to be even if each vertex has even degree. It is known that every even triangulation on any orientable surface with sufficiently large representativity is 4-colorable [J. Hutchinson, B. Richter, P. Seymour, Colouring Eulerian triangulations, J. Combin. Theory, Ser. B 84 (2002) 225-239], but all graphs on any surface with large representativity are 5-colorable [C. Thomassen, Five-coloring maps on surfaces, J. Combin Theory Ser. B 59 (1993) 89-105]. In this paper, we shall characterize 5-chromatic even triangulations with large representativity, which appear only on nonorientable surfaces.     

克莱茵瓶上奇异地图的色数

A map is singular if each edge is on the same face on a surface (i.e., it has only one face on a surface). In this paper we present the chromatic enumeration for rooted singular maps on the Klein bottle.     

DP-4-colorability of planar graphs without intersecting 5-cycles

DP-coloring of graphs as a generalization of list coloring was introduced by Dvo?ák and Postle (2018). In this paper, we show that every planar graph without intersecting 5-cycles is DP-4-colorable, which improves the result of Hu and Wu (2017), who proved that every planar graph without intersecting 5-cycles is 4-choosable, and the results of Kim and Ozeki (2018).     

平面图3色可染的一个充分条件

Steinberg猜想既没有4-圈又没有5-圈的平面图是3色可染的. Xu, Borodin等人各自独立地证明了既没有相邻三角形又没有5-和7-圈的平面图是3 色可染的. 作为这一结果的推论, 没有4-, 5-和7-圈的平面图是3色可染的. 本文证明一个比此推论更接近Steinberg猜想的结果, 设G是一个既没有4-圈又没有5-圈的平面图, 若对每一个k∈{3, 6, 7}, G都不含(k, 7)-弦, 则G是3色可染的, 这里的(k, 7)-弦是指长度为7+k-2的圈的一条弦, 它的两个端点将圈分成两条路, 一条路的长度为6, 另一条路的长度为k-1.     

Note on 3-choosability of planar graphs with maximum degree 4

Deciding whether a planar graph (even of maximum degree 4) is 3-colorable is NP-complete. Determining subclasses of planar graphs being 3-colorable has a long history, but since Grötzsch’s result that triangle-free planar graphs are such, most of the effort was focused to solving Havel’s and Steinberg’s conjectures. In this paper, we prove that every planar graph obtained as a subgraph of the medial graph of any bipartite plane graph is 3-choosable. These graphs are allowed to have close triangles (even incident), and have no short cycles forbidden, hence representing an entirely different class than the graphs inferred by the above mentioned conjectures.     

On minimally b-imperfect graphs

A b-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbour in all other color classes. The b-chromatic number of a graph G is the largest integer k such that G admits a b-coloring with k colors. A graph is b-perfect if the b-chromatic number is equal to the chromatic number for every induced subgraph H of G. A graph is minimally b-imperfect if it is not b-perfect and every proper induced subgraph is b-perfect. We give a list of minimally b-imperfect graphs, conjecture that a graph is b-perfect if and only if it does not contain a graph from this list as an induced subgraph, and prove this conjecture for diamond-free graphs, and graphs with chromatic number at most three.     

The 2-extendability of 5-connected graphs on the Klein bottle

A graph is said to be k-extendable if any independent set of k edges extends to a perfect matching. In this paper, we shall characterize the forbidden structures for 5-connected graphs on the Klein bottle to be 2-extendable. This fact also gives us a sharp lower bound of representativity of 5-connected graphs embedded on the Klein bottle to have such a property, which was considered in Kawarabayashi et al. (submitted for publication) [4].     

3-Trees in polyhedral maps

We show that the vertices of the graph of any polyhedral map on the projective plane, torus or Klein bottle can be covered by a subgraph that is a tree of maximum valence 3. This extends a theorem of the author, who previously proved this theorem for the graphs of 3-dimensional polytopes. Several theorems dealing with paths in polyhedral maps are a consequence of these theorems.     

A note on finite-sheeted covering maps from 2-dimensional compact Abelian groups

We consider finite-sheeted covering maps from 2-dimensional compact connected abelian groups to Klein bottle weak solenoidal spaces, metric continua which are not groups. We show that whenever a group covers a Klein bottle weak solenoidal space it covers groups as well, moreover it covers the product of two solenoids. The converse is not true, we give an example of group which covers groups with any finite number of sheets, but does not cover any Klein bottle weak solenoidal space.     

Uniquely Edge-3-Colorable Graphs and Snarks

 A cubic graph G is uniquely edge-3-colorable if G has precisely one 1-factorization. It is proved in this paper, if a uniquely edge-3-colorable, cubic graph G is cyclically 4-edge-connected, but not cyclically 5-edge-connected, then G must contain a snark as a minor. This is an approach to a conjecture that every triangle free uniquely edge-3-colorable cubic graph must have the Petersen graph as a minor. Fiorini and Wilson (1976) conjectured that every uniquely edge-3-colorable planar cubic graph must have a triangle. It is proved in this paper that every counterexample to this conjecture is cyclically 5-edge-connected and that in a minimal counterexample to the conjecture, every cyclic 5-edge-cut is trivial (an edge-cut T of G is cyclic if no component of G\T is acyclic and a cyclic edge-cut T is trivial if one component of G\T is a circuit of length |T|). Received: July 14, 1997 Revised: June 11, 1998     

没有4至6-圈的平面图是(1,0,0)-可染的

设d₁, d₂,..., d_k 是k个非负整数. 若图G=(V,E) 的顶点集V可剖分成k个子集V₁, V₂,..., V_k,使得对i=1, 2,..., k 由V_i 所导出的子图G[V_i] 的最大度至多为d_i, 则称G是(d₁, d₂,..., d_k)-可染的. 著名的Steinberg 猜想断言, 每个既没有4-圈又没有5-圈的平面图是(0, 0, 0)-可染的. 对此猜想已经证明每个没有4 至7-圈的平面图是(0, 0, 0)-可染的, 但还没有发现有人证明每个没有4 至6-圈的平面图是(0, 0, 0)-可染的. 本文证明没有4 至6-圈的平面图是(1, 0, 0)-可染的.     

Lagrangian Klein Bottles in {\mathbb{R}^{2n}}

It is shown that the n-dimensional Klein bottle admits a Lagrangian embedding into \mathbbR²ⁿ{\mathbb{R}^{2n}} if and only if n is odd.