The width of quadrangulations of the projective plane

We show that every 4‐chromatic graph on n vertices, with no two vertex‐disjoint odd cycles, has an odd cycle of length at most . Let G be a nonbipartite quadrangulation of the projective plane on n vertices. Our result immediately implies that G has edge‐width at most , which is sharp for infinitely many values of n. We also show that G has face‐width (equivalently, contains an odd cycle transversal of cardinality) at most , which is a constant away from the optimal; we prove a lower bound of . Finally, we show that G has an odd cycle transversal of size at most inducing a single edge, where Δ is the maximum degree. This last result partially answers a question of Nakamoto and Ozeki.     

K6‐minors in triangulations and complete quadrangulations

In this paper, we shall prove that a projective‐planar (resp., toroidal) triangulation G has K₆ as a minor if and only if G has no quadrangulation isomorphic to K₄ (resp., K₅ ) as a subgraph. As an application of the theorems, we can prove that Hadwiger's conjecture is true for projective‐planar and toroidal triangulations. © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 302‐312, 2009     

Generating even triangulations of the projective plane

We shall determine the 20 families of irreducible even triangulations of the projective plane. Every even triangulation of the projective plane can be obtained from one of them by a sequence of even‐splittings and attaching octahedra, both of which were first given by Batagelj 2 . © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 333–349, 2007     

Minimal embeddings in the projective plane

We show that if G is a graph embedded on the projective plane in such a way that each noncontractible cycle intersects G at least n times and the embedding is minimal with respect to this property (i.e., the representativity of the embedding is n), then G can be reduced by a series of reduction operations to an n × n × n projective grid. The reduction operations consist of changing a triangle of G to a triad, changing a triad of G to a triangle, and several others. We also show that if every proper minor of the embedding has representativity < n (i.e., the embedding is minimal), then G can be obtained from an n × n × n projective grid by a series of the two reduction operations described above. Hence every minimal embedding has the same number of edges. © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 153–163, 1997     

Einstein hypersurfaces of the Cayley projective plane

We prove that the Cayley hyperbolic plane admits no Einstein hypersurfaces and that the only Einstein hypersurfaces in the Cayley projective plane are geodesic spheres of a certain radius; this completes the classification of Einstein hypersurfaces in rank-one symmetric spaces.     

The Binomial k‐Clique

A finite collection C of k‐sets, where is called a k‐clique if every two k‐sets (called lines) in C have a nonempty intersection and a k‐clique is a called a maximal k‐clique if and C is maximal with respect to this property. That is, every two lines in C have a nonempty intersection and there does not exist A such that , and for all . An elementary example of a maximal k‐clique is furnished by the family of all the k‐subsets of a ‐set. This k‐clique will be called the binomial k‐clique. This paper is intended to give some combinatorial characterizations of the binomial k‐clique as a maximal k‐clique. The techniques developed are then used to provide a large number of examples of mutually nonisomorphic maximal k‐cliques for a fixed value of k.     

Envelopes of splines in the projective plane

In this paper a family of curves—Riemannian cubics—inthe unit sphere and the real projective plane is investigated.Riemannian cubics naturally arise as solutions to variationalproblems in Riemannian spaces. It is remarkable to find thatan envelope of lines generated by a Riemannian cubic in onespace is (nearly) a Riemannian cubic in another space.     

Cayley射影平面的浸入Kaehler曲面

得到了Cayley射影平面的全复浸入Kaehler曲面的一些有趣的拓扑限制。     

The Brownian plane with minimal neck baby universe

For each , let be a uniform rooted quadrangulation, endowed with an appropriate measure, of size n conditioned to have r(n) vertices in its root block. We prove that for a suitable function r(n), after rescaling graph distance by converges to a random pointed non‐compact metric measure space , in the local Gromov‐Hausdorff‐Prokhorov topology. The space is built by identifying a uniform point of the Brownian map with the distinguished point of the Brownian plane. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 729–752, 2017     

Hamiltonian cycles in bipartite quadrangulations on the torus

In this article, we shall prove that every bipartite quadrangulation G on the torus admits a simple closed curve visiting each face and each vertex of G exactly once but crossing no edge. As an application, we conclude that the radial graph of any bipartite quadrangulation on the torus has a hamiltonian cycle. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 69:143‐151, 2012     

Edge proximity and matching extension in projective planar graphs

A graph $G$ with at least $2m+2$ vertices is said to be distance $d$ $m$-extendable if, for any matching $M$ of $G$ with $m$ edges in which the edges lie at distance at least $d$ pairwise, there exists a perfect matching of $G$ containing $M$. In this paper we prove that every 5-connected triangulation on the projective plane of even order is distance 3 7-extendable and distance 4 $m$-extendable for any $m$.     

Facial non‐repetitive edge‐coloring of plane graphs

A sequence r₁, r₂, …, r_2n such that r_i=r_{n+ i} for all 1≤i≤n is called a repetition. A sequence S is called non‐repetitive if no block (i.e. subsequence of consecutive terms of S) is a repetition. Let G be a graph whose edges are colored. A trail is called non‐repetitive if the sequence of colors of its edges is non‐repetitive. If G is a plane graph, a facial non‐repetitive edge‐coloring of G is an edge‐coloring such that any facial trail (i.e. a trail of consecutive edges on the boundary walk of a face) is non‐repetitive. We denote π′_f(G) the minimum number of colors of a facial non‐repetitive edge‐coloring of G. In this article, we show that π′_f(G)≤8 for any plane graph G. We also get better upper bounds for π′_f(G) in the cases when G is a tree, a plane triangulation, a simple 3‐connected plane graph, a hamiltonian plane graph, an outerplanar graph or a Halin graph. The bound 4 for trees is tight. © 2010 Wiley Periodicals, Inc. J Graph Theory 66: 38–48, 2010     

Packing k‐edge trees in graphs of restricted vertex degrees

Let denote the set of graphs with each vertex of degree at least r and at most s, v(G) the number of vertices, and τ_k (G) the maximum number of disjoint k‐edge trees in G. In this paper we show that

• (a1) if G ∈ and s ≥ 4, then τ₂(G) ≥ v(G)/(s + 1),
• (a2) if G ∈ and G has no 5‐vertex components, then τ₂(G) ≥ v(G)4,
• (a3) if G ∈ and G has no k‐vertex component, where k ≥ 2 and s ≥ 3, then τ_k(G) ≥ (v(G) ‐k)/(sk ‐ k + 1), and
• (a4) the above bounds are attained for infinitely many connected graphs.

Our proofs provide polynomial time algorithms for finding the corresponding packings in a graph. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 306–324, 2007     

Edge‐face coloring of plane graphs with maximum degree nine

An edge‐face coloring of a plane graph with edge set E and face set F is a coloring of the elements of E∪F so that adjacent or incident elements receive different colors. Borodin [Discrete Math 128(1–3):21–33, 1994] proved that every plane graph of maximum degree Δ?10 can be edge‐face colored with Δ + 1 colors. We extend Borodin's result to the case where Δ = 9. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:332‐346, 2011     

Embedding of circulant graphs and generalized Petersen graphs on projective plane

Both the circulant graph and the generalized Petersen graph are important types of graphs in graph theory. In this paper, the structures of embeddings of circulant graph C(2n + 1; {1, n}) on the projective plane are described, the number of embeddings of C(2n + 1; {1, n}) on the projective plane follows, then the number of embeddings of the generalized Petersen graph P(2n +1, n) on the projective plane is deduced from that of C(2n +1; {1, n}), because C(2n + 1;{1, n}) is a minor of P(2n + 1, n), their structures of embeddings have relations. In the same way, the number of embeddings of the generalized Petersen graph P(2n, 2) on the projective plane is also obtained.     

A note on k‐shortest paths problem

It is well‐known that in a directed graph, if deleting any edge will not affect the shortest distance between two specific vertices s and t, then there are two edge‐disjoint paths from s to t and both of them are shortest paths. In this article, we generalize this to shortest k edge‐disjoint s‐t paths for any positive integer k. © 2010 Wiley Periodicals, Inc. J Graph Theory 67: 34‐37, 2011     

k‐Kings in k‐Quasitransitive Digraphs

Let D be a digraph with vertex set and arc set . A vertex x is a k‐king of D, if for every , there is an ‐path of length at most k. A subset N of is k‐independent if for every pair of vertices , we have and ; it is l‐absorbent if for every there exists such that . A ‐kernel of D is a k‐independent and l‐absorbent subset of . A k‐kernel is a ‐kernel. A digraph D is k‐quasitransitive, if for any path of length k, x₀ and are adjacent. In this article, we will prove that a k‐quasitransitive digraph with has a k‐king if and only if it has a unique initial strong component and the unique initial strong component is not isomorphic to an extended ‐cycle where each has at least two vertices. Using this fact, we show that every strong k‐quasitransitive digraph has a ‐kernel.