Long cycles in 3‐connected graphs in orientable surfaces

In this article, we apply a cutting theorem of Thomassen to show that there is a function f: N → N such that if G is a 3‐connected graph on n vertices which can be embedded in the orientable surface of genus g with face‐width at least f(g), then G contains a cycle of length at least cn, where c is a constant not dependent on g. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 69–84, 2002     

The wiener index of graphs of arbitrary girth and their line graphs

We consider the invariant W(G) of a simple connected undirected graph G which is equal to the sum of distances between all pairs of its vertices in the natural metric (the Wiener index). We show that, for every g ≥ 5, there is a planar graph G of girth g satisfying W(L(G)) = W(G), where L(G) is the line graph of G.     

On the critical point‐arboricity graphs

In 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 with equality holding iff G contains either K_2k?1 or K_2k?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     

Topological rigidity of algebraic ℙ<Subscript>3</Subscript>-bundles over curves

A projective algebraic surface which is homeomorphic to a ruled surface over a curve of genus g≥1 is itself a ruled surface over a curve of genus g. In this note, we prove the analogous result for projective algebraic manifolds of dimension 4 in the case g≥2. Received: August 30, 2001; in final form: April 12, 2002?Published online: March 12, 2003     

Edge‐partitions of planar graphs and their game coloring numbers

Let G be a planar graph and let g(G) and Δ(G) be its girth and maximum degree, respectively. We show that G has an edge‐partition into a forest and a subgraph H so that (i) Δ(H) ≤ 4 if g(G) ≥ 5; (ii) Δ(H) ≤ 2 if g(G) ≥ 7; (iii) Δ(H)≤ 1 if g(G) ≥ 11; (iv) Δ(H) ≤ 7 if G does not contain 4‐cycles (though it may contain 3‐cycles). These results are applied to find the following upper bounds for the game coloring number col_g(G) of a planar graph G: (i) col_g(G) ≤ 8 if g(G) ≥ 5; (ii) col_g(G)≤ 6 if g(G) ≥ 7; (iii) col_g(G) ≤ 5 if g(G) ≥ 11; (iv) col_g(G) ≤ 11 if G does not contain 4‐cycles (though it may contain 3‐cycles). © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 307–317, 2002     

Circular flows of nearly Eulerian graphs and vertex‐splitting

The odd edge connectivity of a graph G, denoted by λ_o(G), is the size of a smallest odd edge cut of the graph. Let S be any given surface and ? be a positive real number. We proved that there is a function f_S(?) (depends on the surface S and lim_?→0 f_S(?)=∞) such that any graph G embedded in S with the odd‐edge connectivity at least f_S(?) admits a nowhere‐zero circular (2+?)‐flow. Another major result of the work is a new vertex splitting lemma which maintains the old edge connectivity and graph embedding. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 147–161, 2002     

Maximizing Algebraic Connectivity Over Unicyclic Graphs

We consider the class of unicyclic graphs on n vertices with girth g, and over that class, we attempt to determine which graph maximizes the algebraic connectivity. When g is fixed, we show that there is an N such that for each n>N, the maximizing graph consists of a g cycle with n?g pendant vertices adjacent to a common vertex on the cycle. We also provide a bound on N. On the other hand, when g is large relative to n, we show that this graph does not maximize the algebraic connectivity, and we give a partial discussion of the nature of the maximizing graph in that situation.     

The maximum interval number of graphs with given genus

The interval number of a graph G, denoted i(G), is the least positive integer t such that G is the intersection graph of sets, each of which is the union of t compact real intervals. It is known that every planar graph has interval number at most 3 and that this result is best possible. We investigate the maximum value of the interval number for graphs with higher genus and show that the maximum value of the interval number of graphs with genus g is between ?√g? and 3 + ?√3g?. We also show that the maximum arboricity of graphs with genus g is either 1 + ?√3g? or 2 + ?√3g?.     

Uniqueness and minimality of large face-width embeddings of graphs

LetG be a graph embedded in a surface of genusg. It is shown that if the face-width of the embedding is at leastclog(g)/loglog(g), then such an embedding is unique up to Whitney equivalence. If the face-width is at leastclog(g), then every embedding ofG which is not Whitney equivalent to our embedding has strictly smaller Euler characteristic.Supported in part by the Ministry of Science and Technology of Slovenia, Research Project P1-0210-101-94.     

Linear coloring of graphs embeddable in a surface of nonnegative characteristic

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G. In this paper, we prove that every graph G with girth g(G) and maximum degree Δ(G) that can be embedded in a surface of nonnegative characteristic has lc(G) = Δ(2G )+ 1 if there is a pair (Δ, g) ∈ {(13, 7), (9, 8), (7, 9), (5, 10), (3, 13)} such that G s...     

The maximum valency of regular graphs with given order and odd girth

The odd girth of a graph G is the length of a shortest odd cycle in G. Let d(n, g) denote the largest k such that there exists a k-regular graph of order n and odd girth g. It is shown that dn, g ≥ 2|n/g≥ if n ≥ 2g. As a consequence, we prove a conjecture of Pullman and Wormald, which says that there exists a 2j-regular graph of order n and odd girth g if and only if n ≥ gj, where g ≥ 5 is odd and j ≥ 2. A different variation of the problem is also discussed.     

Primal separation for 0/1 polytopes

 The 0/1 primal separation problem is: Given an extreme point xˉ of a 0/1 polytope P and some point x ^*, find an inequality which is tight at xˉ, violated by x ^* and valid for P or assert that no such inequality exists. It is known that this separation variant can be reduced to the standard separation problem for P. We show that 0/1 optimization and 0/1 primal separation are polynomial time equivalent. This implies that the problems 0/1 optimization, 0/1 standard separation, 0/1 augmentation, and 0/1 primal separation are polynomial time equivalent. Then we provide polynomial time primal separation procedures for matching, stable set, maximum cut, and maximum bipartite graph problems, giving evidence that these algorithms are conceptually simpler and easier to implement than their corresponding counterparts for standard separation. In particular, for perfect matching we present an algorithm for primal separation that rests only on simple max-flow computations. In contrast, the known standard separation method relies on an explicit minimum odd cut algorithm. Consequently, we obtain a very simple proof that a maximum weight perfect matching of a graph can be computed in polynomial time. Received: August 20, 2001 / Accepted: April 2002 Published online: December 9, 2002 RID="⋆" ID="⋆" This research was developed while the author was on leave at the Istituto di Analisi dei Sistemi ed Informatica, Viale Manzoni 30, 00185 Roma, supported by the project TMR-DONET nr. ERB FMRX-CT98-0202 of the European Union. Mathematics Subject Classification (2000): 90C10, 90C60, 90C57     

Geodesic Laminations Revisited

The Bratteli diagram is an infinite graph which reflects the structure of projections in an AF-algebra. We prove that every strictly ergodic unimodular Bratteli diagram of rank 2g+m−1 gives rise to a minimal geodesic lamination with the m principal regions on a hyperbolic surface of genus g≥1. The proof is based on a Morse theory of the recurrent geodesics on the hyperbolic surfaces.     

2‐ and 3‐factors of graphs on surfaces

It has been conjectured that any 5‐connected graph embedded in a surface Σ with sufficiently large face‐width is hamiltonian. This conjecture was verified by Yu for the triangulation case, but it is still open in general. The conjecture is not true for 4‐connected graphs. In this article, we shall study the existence of 2‐ and 3‐factors in a graph embedded in a surface Σ. A hamiltonian cycle is a special case of a 2‐factor. Thus, it is quite natural to consider the existence of these factors. We give an evidence to the conjecture in a sense of the existence of a 2‐factor. In fact, we only need the 4‐connectivity with minimum degree at least 5. In addition, our face‐width condition is not huge. Specifically, we prove the following two results. Let G be a graph embedded in a surface Σ of Euler genus g.

• (1) If G is 4‐connected and minimum degree of G is at least 5, and furthermore, face‐width of G is at least 4g?12, then G has a 2‐factor.
• (2) If G is 5‐connected and face‐width of G is at least max{44g?117, 5}, then G has a 3‐factor.

The connectivity condition for both results are best possible. In addition, the face‐width conditions are necessary too. Copyright © 2010 Wiley Periodicals, Inc. J Graph Theory 67:306‐315, 2011     

Connected Factors and Spanning Trees in Graphs

 Let G be a graph, and g, f, f′ be positive integer-valued functions defined on V(G). If an f′-factor of G is a spanning tree, we say that it is f′-tree. In this paper, it is shown that G contains a connected (g, f+f′−1)-factor if G has a (g, f)-factor and an f′-tree. Received: October 30, 2000 Final version received: August 20, 2002     

Optimally Cutting a Surface into a Disk

We consider the problem of cutting a subset of the edges of a polyhedral manifold surface, possibly with boundary, to obtain a single topological disk, minimizing either the total number of cut edges or their total length. We show that this problem is NP-hard in general, even for manifolds without boundary and for punctured spheres. We also describe an algorithm with running time n ^O(g+k), where n is the combinatorial complexity, g is the genus, and k is the number of boundary components of the input surface. Finally, we describe a greedy algorithm that outputs a O(log² g)-approximation of the minimum cut graph in O(g ² n log n) time.     

A lower bound on the order of regular graphs with given girth pair

The girth pair of a graph gives the length of a shortest odd and a shortest even cycle. The existence of regular graphs with given degree and girth pair was proved by Harary and Kovács [Regular graphs with given girth pair, J Graph Theory 7 ( 1 ), 209–218]. A (δ, g)‐cage is a smallest δ‐regular graph with girth g. For all δ ≥ 3 and odd girth g ≥ 5, Harary and Kovács conjectured the existence of a (δ,g)‐cage that contains a cycle of length g + 1. In the main theorem of this article we present a lower bound on the order of a δ‐regular graph with odd girth g ≥ 5 and even girth h ≥ g + 3. We use this bound to show that every (δ,g)‐cage with δ ≥ 3 and g ∈ {5,7} contains a cycle of length g + 1, a result that can be seen as an extension of the aforementioned conjecture by Harary and Kovács for these values of δ, g. Moreover, for every odd g ≥ 5, we prove that the even girth of all (δ,g)‐cages with δ large enough is at most (3g ? 3)/2. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 153–163, 2007     

Stability of quasi-geodesics in Teichmüller space

Let S be a surface S of genus g ≥ 0 with m ≥ 0 punctures and 3g − 3 + m ≥ 2. We show that a Teichmüller quasi-geodesic in the thick part of Teichmüller space for S is contained in a bounded neighborhood of a geodesic if and only if it induces a quasi-geodesic in the curve graph.     

The geodetic number of the lexicographic product of graphs

A set S of vertices of a graph G is a geodetic set if every vertex of G lies in an interval between two vertices from S. The size of a minimum geodetic set in G is the geodetic number g(G) of G. We find that the geodetic number of the lexicographic product G°H for a non-complete graph H lies between 2 and 3g(G). We characterize the graphs G and H for which g(G°H)=2, as well as the lexicographic products T°H that enjoy g(T°H)=3g(G), when T is isomorphic to a tree. Using a new concept of the so-called geodominating triple of a graph G, a formula that expresses the exact geodetic number of G°H is established, where G is an arbitrary graph and H a non-complete graph.