Six-Critical Graphs on the Klein Bottle

We exhibit an explicit list of nine graphs such that a graph drawn in the Klein bottle is 5-colorable if and only if it has no subgraph isomorphic to a member of the list. This answers a question of Thomassen [J. Comb. Theory Ser. B 70 (1997), 67–100] and implies an earlier result of Král', Mohar, Nakamoto, Pangrác and Suzuki that an Eulerian triangulation of the Klein bottle is 5-colorable if and only if it has no complete subgraph on six vertices.     

Grünbaum colorings of even triangulations on surfaces

A Grünbaum coloring of a triangulation G is a map c : such that for each face f of G, the three edges of the boundary walk of f are colored by three distinct colors. By Four Color Theorem, it is known that every triangulation on the sphere has a Grünbaum coloring. So, in this article, we investigate the question whether each even (i.e., Eulerian) triangulation on a surface with representativity at least r has a Grünbaum coloring. We prove that, regardless of the representativity, every even triangulation on a surface has a Grünbaum coloring as long as is the projective plane, the torus, or the Klein bottle, and we observe that the same holds for any surface with sufficiently large representativity. On the other hand, we construct even triangulations with no Grünbaum coloring and representativity , and 3 for all but finitely many surfaces. In dual terms, our results imply that no snark admits an even map on the projective plane, the torus, or the Klein bottle, and that all but finitely many surfaces admit an even map of a snark with representativity at least 3.     

3-Colorable Even Embeddings on Closed Surfaces

It is known that for any closed surface F², every even embedding on F² with sufficiently large representativity is 4-colorable. In this paper, we shall characterize 3-colorable even embeddings on F² with sufficiently large representativity.     

On 3-choosable planar graphs of girth at least 4

Let G be a plane graph of girth at least 4. Two cycles of G are intersecting if they have at least one vertex in common. In this paper, we show that if a plane graph G has neither intersecting 4-cycles nor a 5-cycle intersecting with any 4-cycle, then G is 3-choosable, which extends one of Thomassen’s results [C. Thomassen, 3-list-coloring planar graphs of girth 5, J. Combin. Theory Ser. B 64 (1995) 101-107].     

The multivariate signed Bollobás-Riordan polynomial

We generalise the signed Bollobás-Riordan polynomial of S. Chmutov and I. Pak [S. Chmutov, I. Pak, The Kauffman bracket of virtual links and the Bollobás-Riordan polynomial, Mos. Math. J. 7(3) (2007), 409-418] to a multivariate signed polynomial Z and study its properties. We prove the invariance of Z under the recently defined partial duality of S. Chmutov [S. Chmutov, Generalized duality for graphs on surfaces and the signed Bollobás-Riordan polynomial, J. Combin. Theory, Ser. B 99(3) (2009), 617-638. arXiv:0711.3490, doi:10.1016/j.jctb.2008.09.007] and show that the duality transformation of the multivariate Tutte polynomial is a direct consequence of it.     

Signed differential posets and sign-imbalance

We study signed differential posets, a signed version of differential posets. These posets satisfy enumerative identities which are signed analogues of those satisfied by differential posets. Our main motivations are the sign-imbalance identities for partition shapes originally conjectured by Stanley, now proven in [T. Lam, Growth diagrams, domino insertion and sign-imbalance, J. Combin. Theory Ser. A 107 (2004) 87-115; A. Reifergerste, Permutation sign under the Robinson-Schensted-Knuth correspondence, Ann. Comb. 8 (2004) 103-112; J. Sjöstrand, On the sign-imbalance of partition shapes, J. Combin. Theory Ser. A 111 (2005) 190-203]. We show that these identities result from a signed differential poset structure on Young's lattice, and explain similar identities for Fibonacci shapes.     

Bordeaux 3-color conjecture and 3-choosability

A graph G=(V,E) is list L-colorable if for a given list assignment L={L(v):v∈V}, there exists a proper coloring c of G such that c(v)∈L(v) for all v∈V. If G is list L-colorable for every list assignment with |L(v)|?k for all v∈V, then G is said to be k-choosable.In this paper, we prove that (1) every planar graph either without 4- and 5-cycles, and without triangles at distance less than 4, or without 4-, 5- and 6-cycles, and without triangles at distance less than 3 is 3-choosable; (2) there exists a non-3-choosable planar graph without 4-cycles, 5-cycles, and intersecting triangles. These results have some consequences on the Bordeaux 3-color conjecture by Borodin and Raspaud [A sufficient condition for planar graphs to be 3-colorable. J. Combin. Theory Ser. B 88 (2003) 17-27].     

N‐flips in even triangulations on the sphere

A triangulation is said to be even if each vertex has even degree. For even triangulations, define the N‐flip and the P2‐flip as two deformations preserving the number of vertices. We shall prove that any two even triangulations on the sphere with the same number of vertices can be transformed into each other by a sequence of N‐ and P2‐flips. © 2005 Wiley Periodicals, Inc. J. Graph Theory     

Extending Greene's theorem to directed graphs

The purpose of this note is to prove a theorem on acyclic digraphs, conjectured by Linial (J. Combin. Theory Ser. A, 30 (1981), 331–334), which includes Greene's theorem (J. Combin. Theory Ser. A, 20 (1976), 69–79) on decompositions of a partially ordered set into antichains.     

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     

A new short proof of a theorem of Ahlswede and Khachatrian

Ahlswede and Khachatrian [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121-138] proved the following theorem, which answered a question of Frankl and Füredi [P. Frankl, Z. Füredi, Nontrivial intersecting families, J. Combin. Theory Ser. A 41 (1986) 150-153]. Let 2?t+1?k?2t+1 and n?(t+1)(k−t+1). Suppose that F is a family of k-subsets of an n-set, every two of which have at least t common elements. If |_?F∈FF|<t, then , and this is best possible. We give a new, short proof of this result. The proof in [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121-138] requires the entire machinery of the proof of the complete intersection theorem, while our proof uses only ordinary compression and an earlier result of Wilson [R.M. Wilson, The exact bound in the Erd?s-Ko-Rado theorem, Combinatorica 4 (1984) 247-257].     

On semi-Eulerian partially ordered sets with boundary

Eulerian posets are motivated by the posets from triangulations of spheres; semi-Eulerian posets are motivated by the posets from triangulations of manifolds. Motivated by investigation (Proc. Natl. Acad. Sci. USA 95 (1998) 9093; Adv. Appl. Math. 19 (1997) 144; J. Combin. Theory Ser. A 85 (1999) 1; Adv. Appl. Math. 21 (1998) 22) on the number of faces of triangulations of manifolds with boundary, we introduce semi-Eulerian posets with boundary in this paper, and generalize the reciprocity laws, the Dehn–Sommerville equations, and the combinatorial Alexander duality to semi-Eulerian posets with boundary.     

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.     

On the strong p-Helly property

The notion of strong p-Helly hypergraphs was introduced by Golumbic and Jamison in 1985 [M.C. Golumbic, R.E. Jamison, The edge intersection graphs of paths in a tree, J. Combin. Theory Ser. B 38 (1985) 8-22]. Independently, other authors [A. Bretto, S. Ubéda, J. ?erovnik, A polynomial algorithm for the strong Helly property. Inform. Process. Lett. 81 (2002) 55-57, E. Prisner, Hereditary clique-Helly graphs, J. Combin. Math. Combin. Comput. 14 (1993) 216-220, W.D. Wallis, Guo-Hui Zhang, On maximal clique irreducible graphs. J. Combin. Math. Combin. Comput. 8 (1990) 187-193.] have also considered the strong Helly property in other contexts. In this paper, we characterize strong p-Helly hypergraphs. This characterization leads to an algorithm for recognizing such hypergraphs, which terminates within polynomial time whenever p is fixed. In contrast, we show that the recognition problem is co-NP-complete, for arbitrary p. Further, we apply the concept of strong p-Helly hypergraphs to the cliques of a graph, leading to the class of strong p-clique-Helly graphs. For p=2, this class is equivalent to that of hereditary clique-Helly graphs [E. Prisner, Hereditary clique-Helly graphs, J. Combin. Math. Combin. Comput. 14 (1993) 216-220]. We describe a characterization for this class and obtain an algorithm for recognizing such graphs. Again, the algorithm has polynomial-time complexity for p fixed, and we show the corresponding recognition problem to be NP-hard, for arbitrary p.     

On an upper bound of the spectral radius of graphs

We give some upper bounds for the spectral radius of bipartite graph and graph, which improve the result in Hong’s Paper [Y. Hong, J.-L. Shu, K. Fang, A sharp upper bound of the spectral radius of graphs, J. Combin. Theory Ser. B 81 (2001) 177-183].     

On graphs whose square have strong hamiltonian properties

The squareG² of a graph G is the graph having the same vertex set as G and two vertices are adjacent if and only if they are at distance at most 2 from each other. It is known that if G has no cut-vertex, then G² is Hamilton-connected (see [G. Chartrand, A.M. Hobbs, H.A. Jung, S.F. Kapoor, C.St.J.A. Nash-Williams, The square of a block is hamiltonian connected, J. Combin. Theory Ser. B 16 (1974) 290-292; R.J. Faudree and R.H. Schelp, The square of a block is strongly path connected, J. Combin. Theory Ser. B 20 (1976) 47-61]). We prove that if G has only one cut-vertex, then G² is Hamilton-connected. In the case that G has only two cut-vertices, we prove that if the block that contains the two cut-vertices is hamiltonian, then G² is Hamilton-connected. Further, we characterize all graphs with at most one cycle having Hamilton-connected square.     

The graph of perfect matching polytope and an extreme problem

For graph G, its perfect matching polytope Poly(G) is the convex hull of incidence vectors of perfect matchings of G. The graph corresponding to the skeleton of Poly(G) is called the perfect matching graph of G, and denoted by PM(G). It is known that PM(G) is either a hypercube or hamilton connected [D.J. Naddef, W.R. Pulleyblank, Hamiltonicity and combinatorial polyhedra, J. Combin. Theory Ser. B 31 (1981) 297-312; D.J. Naddef, W.R. Pulleyblank, Hamiltonicity in (0-1)-polytope, J. Combin. Theory Ser. B 37 (1984) 41-52]. In this paper, we give a sharp upper bound of the number of lines for the graphs G whose PM(G) is bipartite in terms of sizes of elementary components of G and the order of G, respectively. Moreover, the corresponding extremal graphs are constructed.     

Cycles and perfect matchings

Fan Chung and Ron Graham (J. Combin. Theory Ser. B 65 (1995) 273–290) introduced the cover polynomial for a directed graph and showed that it was connected with classical rook theory. Dworkin (J. Combin. Theory Ser. B 71 (1997) 17–53) showed that the cover polynomial naturally factors for directed graphs associated with Ferrers boards. The authors (Adv. Appl. Math. 27 (2001) 438–481) developed a rook theory for shifted Ferrers boards where the analogue of a rook placement is replaced by a partial perfect matching of K_2n, the complete graph on 2n vertices. In this paper, we show that an analogue of Dworkin's result holds for shifted Ferrers boards in this setting. We also show how cycle-counting matching numbers are connected to cycle-counting “hit numbers” (which involve perfect matchings of K_2n).     

Disjoint paths, planarizing cycles, and spanning walks

We study the existence of certain disjoint paths in planar graphs and generalize a theorem of Thomassen on planarizing cycles in surfaces. Results are used to prove that every 5-connected triangulation of a surface with sufficiently large representativity is hamiltonian, thus verifying a conjecture of Thomassen. We also obtain results about spanning walks in graphs embedded in a surface with large representativity.