The shifted Turán sieve method on tournaments II

In a previous work [5], we developed the shifted Turán sieve method on a bipartite graph and applied it to problems on cycles in tournaments. More precisely, we obtained upper bounds for the number of tournaments which contain a small number of r-cycles. In this paper, we improve our sieve inequality and apply it to obtain an upper bound for the number of bipartite tournaments which contain a number of 2r-cycles far from the average. We also provide the exact bound for the number of tournaments which contain few 3-cycles, using other combinatorial arguments.     

Chordal bipartite completion of colored graphs

Golumbic, Kaplan, and Shamir [Graph sandwich problems, J. Algorithms 19 (1995) 449-473], in their paper on graph sandwich problems published in 1995, left the status of the sandwich problems for strongly chordal graphs and chordal bipartite graphs open. It was recently shown [C.M.H. de Figueiredo, L. Faria, S. Klein, R. Sritharan, On the complexity of the sandwich problems for strongly chordal graphs and chordal bipartite graphs, Theoret. Comput. Sci., accepted for publication] that the sandwich problem for strongly chordal graphs is NP-complete. We show that given graph G with a proper vertex coloring c, determining whether there is a supergraph of G that is chordal bipartite and also is properly colored by c is NP-complete. This implies that the sandwich problem for chordal bipartite graphs is also NP-complete.     

Counting substructures I: Color critical graphs

Let F be a graph which contains an edge whose deletion reduces its chromatic number. We prove tight bounds on the number of copies of F in a graph with a prescribed number of vertices and edges. Our results extend those of Simonovits (1968) [8], who proved that there is one copy of F, and of Rademacher, Erd?s (1962) [1] and [2] and Lovász and Simonovits (1983) [4], who proved similar counting results when F is a complete graph.One of the simplest cases of our theorem is the following new result. There is an absolute positive constant c such that if n is sufficiently large and 1?q<cn, then every n vertex graph with ⌊n²/4⌋+q edges contains at least

     

Enumerating spanning trees of graphs with an involution

As the extension of the previous work by Ciucu and the present authors [M. Ciucu, W.G. Yan, F.J. Zhang, The number of spanning trees of plane graphs with reflective symmetry, J. Combin. Theory Ser. A 112 (2005) 105-116], this paper considers the problem of enumeration of spanning trees of weighted graphs with an involution which allows fixed points. We show that if G is a weighted graph with an involution, then the sum of weights of spanning trees of G can be expressed in terms of the product of the sums of weights of spanning trees of two weighted graphs with a smaller size determined by the involution of G. As applications, we enumerate spanning trees of the almost-complete bipartite graph, the almost-complete graph, the Möbius ladder, and the almost-join of two copies of a graph.     

Super restricted edge connectivity of regular graphs with two orbits

It is well known that the edge-connectivity of a simple, connected, vertex transitive graph attains its regular degree. It is then natural to consider the relationship between the graph’s edge connectivity and the number of orbits of its automorphism group. In [6], Liu and Meng (2008) studied the edge connectivity of regular double-orbits graphs. Later, Lin et al. (in press) [10] characterized the λ′-optimal 3-regular double-orbit graph and given a sufficient condition for the k-regular double-orbit graphs to be optimal. In this note, we characterize the super restricted edge connected k-regular double-orbit graphs with grith at least 6.     

An upper bound for the competition numbers of graphs

A hole of a graph is an induced cycle of length at least 4. Kim (2005) [2] conjectured that the competition number k(G) is bounded by h(G)+1 for any graph G, where h(G) is the number of holes of G. In Lee et al. [3], it is proved that the conjecture is true for a graph whose holes are mutually edge-disjoint. In Li et al. (2009) [4], it is proved that the conjecture is true for a graph, all of whose holes are independent. In this paper, we prove that Kim’s conjecture is true for a graph G satisfying the following condition: for each hole C of G, there exists an edge which is contained only in C among all induced cycles of G.     

Invalid proofs on incidence coloring

Incidence coloring of a graph G is a mapping from the set of incidences to a color-set C such that adjacent incidences of G are assigned distinct colors. Since 1993, numerous fruitful results as regards incidence coloring have been proved. However, some of them are incorrect. We remedy the error of the proof in [R.A. Brualdi, J.J.Q. Massey, Incidence and strong edge colorings of graphs, Discrete Math. 122 (1993) 51-58] concerning complete bipartite graphs. Also, we give an example to show that an outerplanar graph with Δ=4 is not 5-incidence colorable, which contradicts [S.D. Wang, D.L. Chen, S.C. Pang, The incidence coloring number of Halin graphs and outerplanar graphs, Discrete Math. 256 (2002) 397-405], and prove that the incidence chromatic number of the outerplanar graph with Δ≥7 is Δ+1. Moreover, we prove that the incidence chromatic number of the cubic Halin graph is 5. Finally, to improve the lower bound of the incidence chromatic number, we give some sufficient conditions for graphs that cannot be (Δ+1)-incidence colorable.     

The geometry of graphs and some of its algorithmic applications

In this paper we explore some implications of viewing graphs asgeometric objects. This approach offers a new perspective on a number of graph-theoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that respect themetric of the (possibly weighted) graph. Given a graphG we map its vertices to a normed space in an attempt to (i) keep down the dimension of the host space, and (ii) guarantee a smalldistortion, i.e., make sure that distances between vertices inG closely match the distances between their geometric images. In this paper we develop efficient algorithms for embedding graphs low-dimensionally with a small distortion. Further algorithmic applications include:

1. A simple, unified approach to a number of problems on multicommodity flows, including the Leighton-Rao Theorem [37] and some of its extensions. We solve an open question in this area, showing that the max-flow vs. min-cut gap in thek-commodities problem isO(logk). Our new deterministic polynomial-time algorithm finds a (nearly tight) cut meeting this bound.
2. For graphs embeddable in low-dimensional spaces with a small distortion, we can find low-diameter decompositions (in the sense of [7] and [43]). The parameters of the decomposition depend only on the dimension and the distortion and not on the size of the graph.
3. In graphs embedded this way, small balancedseparators can be found efficiently.

Given faithful low-dimensional representations of statistical data, it is possible to obtain meaningful and efficientclustering. This is one of the most basic tasks in pattern-recognition. For the (mostly heuristic) methods used in the practice of pattern-recognition, see [20], especially chapter 6. Our studies of multicommodity flows also imply that every embedding of (the metric of) ann-vertex, constant-degree expander into a Euclidean space (of any dimension) has distortion Ω(logn). This result is tight, and closes a gap left open by Bourgain [12].     

Cohesion and stability in graphs

The cohesion of a vertex v is the minimum number of edges whose deletion makes v a cutvertex of the resulting graph. Such considerations are particularly interesting for alliance and friendship graphs. In this paper, we further examine cohesion, first introduced in [3], with our major goal here being the examination of stability under edge addition. We find necessary and sufficient conditions for a vertex to be stable and prove a theorem which discovers total vertex stability in several graphs.     

Local chromatic number of quadrangulations of surfaces

The local chromatic number of a graph G, as introduced in [4], is the minimum integer k such that G admits a proper coloring (with an arbitrary number of colors) in which the neighborhood of each vertex uses less than k colors. In [17] a connection of the local chromatic number to topological properties of (a box complex of) the graph was established and in [18] it was shown that a topological condition implying the usual chromatic number being at least four has the stronger consequence that the local chromatic number is also at least four. As a consequence one obtains a generalization of the following theorem of Youngs [19]: If a quadrangulation of the projective plane is not bipartite it has chromatic number four. The generalization states that in this case the local chromatic number is also four. Both papers [1] and [13] generalize Youngs’ result to arbitrary non-orientable surfaces replacing the condition of the graph being not bipartite by a more technical condition of an odd quadrangulation. This paper investigates when these general results are true for the local chromatic number instead of the chromatic number. Surprisingly, we find out that (unlike in the case of the chromatic number) this depends on the genus of the surface. For the non-orientable surfaces of genus at most four, the local chromatic number of any odd quadrangulation is at least four, but this is not true for non-orientable surfaces of genus 5 or higher. We also prove that face subdivisions of odd quadrangulations and Fisk triangulations of arbitrary surfaces exhibit the same behavior for the local chromatic number as they do for the usual chromatic number.     

On the minimal energy of unicyclic Hückel molecular graphs possessing Kekulé structures

Let G be any unicyclic Hückel molecular graph with Kekulé structures on n vertices where n≥8 is an even number. In [W. Wang, A. Chang, L. Zhang, D. Lu, Unicyclic Hückel molecular graphs with minimal energy, J. Math. Chem. 39 (1) (2006) 231-241], Wang et al. showed that if G satisfies certain conditions, then the energy of G is always greater than the energy of the radialene graph. In this paper we prove that this inequality actually holds under a much weaker condition.     

n≤12阶(k,l)-正则极大平面图

我们知道当图的顶点数n>12时不存在正则极大平面图.相关文献提出了(k,l)-正则极大平面图的概念,并讨论了(5,6)-正则极大平面图的存在性.在相关文献中,作者分别讨论了阶n>12的(k,l)-正则极大平面图的存在条件及构造方法.本文讨论了阶n(≤12)的(k,l)-正则极大平面图的存在性,除两种情况外,本文给出了阶n(≤12)的(k,l)-正则极大平面图的存在条件及其一种构造的例子.     

On P-quasigroups and decompositions of complete undirected graphs

In [2], A. Kotzig has introduced the concepts of P-groupoid and P-quasigroup and has shown how these concepts are related to the decomposition of a complete undirected graph into disjoint closed paths. To each closed path of the graph associated with a given P-quasigroup Q there corresponds a cyclic partial transversal in the Latin square L which is defined by the multiplication table of Q. In this paper, it is shown that cyclic transversals are connected with Hamiltonian decompositions of complete undirected graphs having an even number of vertices and a connection between the order of a particular type of P-quasigroup and the length of its cyclic partial transversals is established. An indirect connection with the work of Yap [4] is established via the concept of isotopy.     

Total chromatic number of graphs with small genus

Given a graph G, a total k-coloring of G is a simultaneous coloring of the vertices and edges of G with k colors. Denote χ_ve (G) the total chromatic number of G, and c(Σ) the Euler characteristic of a surfase Σ. In this paper, we prove that for any simple graph G which can be embedded in a surface Σ with Euler characteristic c(Σ), χ_ve (G) = Δ (G) + 1 if c(Σ) > 0 and Δ (G) ≥ 13, or, if c(Σ) = 0 and Δ (G) ≥ 14. This result generalizes results in [3], [4], [5] by Borodin.     

A note on the critical ideals of a cycle

Critical ideals of a graph G which are the determinantal ideals of its generalized Laplacian matrix were first introduced by Corrales and Valencia as a generalization of the critical group. Then it was shown that critical ideals are also closely related to other properties of the graph, such as the clique number and the zero forcing number. In this note, we give a simple proof of Theorem 4.13 proved in [7], which gives a Gröbner basis of the first nontrivial critical ideal of a cycle. After that as applications we determine explicit expressions for the characteristic ideals of a cycle and the critical groups of a family of thick wheels.     

Improper colouring of (random) unit disk graphs

For any graph G, the k-improper chromatic numberχ^k(G) is the smallest number of colours used in a colouring of G such that each colour class induces a subgraph of maximum degree k. We investigate χ^k for unit disk graphs and random unit disk graphs to generalise results of McDiarmid and Reed [Colouring proximity graphs in the plane, Discrete Math. 199(1-3) (1999) 123-137], McDiarmid [Random channel assignment in the plane, Random Structures Algorithms 22(2) (2003) 187-212], and McDiarmid and Müller [On the chromatic number of random geometric graphs, submitted for publication].     

Vertex splitting and the recognition of trapezoid graphs

Trapezoid graphs are the intersection family of trapezoids where every trapezoid has a pair of opposite sides lying on two parallel lines. These graphs have received considerable attention and lie strictly between permutation graphs (where the trapezoids are lines) and cocomparability graphs (the complement has a transitive orientation). The operation of “vertex splitting”, introduced in (Cheah and Corneil, 1996) [3], first augments a given graph G and then transforms the augmented graph by replacing each of the original graph’s vertices by a pair of new vertices. This “splitted graph” is a permutation graph with special properties if and only if G is a trapezoid graph. Recently vertex splitting has been used to show that the recognition problems for both tolerance and bounded tolerance graphs is NP-complete (Mertzios et al., 2010) [11]. Unfortunately, the vertex splitting trapezoid graph recognition algorithm presented in (Cheah and Corneil, 1996) [3] is not correct. In this paper, we present a new way of augmenting the given graph and using vertex splitting such that the resulting algorithm is simpler and faster than the one reported in (Cheah and Corneil, 1996) [3].     

A note on power domination in grid graphs

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well known vertex covering and dominating set problems in graphs (see [T.W. Haynes, S.M. Hedetniemi, S.T. Hedetniemi, M.A. Henning, Power domination in graphs applied to electrical power networks, SIAM J. Discrete Math. 15(4) (2002) 519-529]). A set S of vertices is defined to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set S (following a set of rules for power system monitoring). The minimum cardinality of a power dominating set of a graph is its power domination number. In this paper, we determine the power domination number of an n×m grid graph.     

Constructing Ramsey graphs from Boolean function representations

Explicit construction of Ramsey graphs or graphs with no large clique or independent set, has remained a challenging open problem for a long time. While Erdös’ probabilistic argument shows the existence of graphs on 2n vertices with no clique or independent set of size 2 ⁿ , the best explicit constructions achieve a far weaker bound. There is a connection between Ramsey graph constructions and polynomial representations of Boolean functions due to Grolmusz; a low degree representation for the OR function can be used to explicitly construct Ramsey graphs [17,18]. We generalize the above relation by proposing a new framework. We propose a new definition of OR representations: a pair of polynomials represent the OR function if the union of their zero sets contains all points in {0, 1} ⁿ except the origin. We give a simple construction of a Ramsey graph using such polynomials. Furthermore, we show that all the known algebraic constructions, ones to due to Frankl-Wilson [12], Grolmusz [18] and Alon [2] are captured by this framework; they can all be derived from various OR representations of degree O(√n) based on symmetric polynomials. Thus the barrier to better Ramsey constructions through such algebraic methods appears to be the construction of lower degree representations. Using new algebraic techniques, we show that better bounds cannot be obtained using symmetric polynomials.     

Precoloring extension on unit interval graphs

In the precoloring extension problem a graph is given with some of the vertices having preassigned colors and it has to be decided whether this coloring can be extended to a proper k-coloring of the graph. Answering an open question of Hujter and Tuza [Precoloring extension. III. Classes of perfect graphs, Combin. Probab. Comput. 5 (1) (1996) 35-56], we show that the precoloring extension problem is NP-complete on unit interval graphs.