On Walkup's class K (d) and a minimal triangulation of (S-3 (sic) S-1)(#3)

For d >= 2, Walkup's class K (d) consists of the d-dimensional simplicial complexes all whose vertex-links are stacked (d - 1)-spheres. Kalai showed that for d >= 4, all connected members of K (d) are obtained from stacked d-spheres by finitely many elementary handle additions. According to a result of Walkup, the face vector of any triangulated 4-manifold X with Euler characteristic chi satisfies f(1) >= 5f(0) - 15/2 chi, with equality only for X is an element of K(4). Kuhnel observed that this implies f(0)(f(0) - 11) >= -15 chi, with equality only for 2-neighborly members of K(4). Kuhnel also asked if there is a triangulated 4-manifold with f(0) = 15, chi = -4 (attaining equality in his lower bound). In this paper, guided by Kalai's theorem, we show that indeed there is such a triangulation. It triangulates the connected sum of three copies of the twisted sphere product S-3 (sic) S-1. Because of Kuhnel's inequality, the given triangulation of this manifold is a vertex-minimal triangulation. By a recent result of Effenberger, the triangulation constructed here is tight. Apart from the neighborly 2-manifolds and the infinite family of (2d + 3)-vertex sphere products Sd-1 X S-1 (twisted for d odd), only fourteen tight triangulated manifolds were known so far. The present construction yields a new member of this sporadic family. We also present a self-contained proof of Kalai's result. (C) 2011 Elsevier B.V. All rights reserved.

Acyclic Edge-Coloring of Planar Graphs

A proper edge-coloring with the property that every cycle contains edges of at least three distinct colors is called an acyclic edge-coloring. The acyclic chromatic index of a graph G, denoted. chi'(alpha)(G), is the minimum k such that G admits an acyclic edge-coloring with k colors. We conjecture that if G is planar and Delta(G) is large enough, then chi'(alpha) (G) = Delta (G). We settle this conjecture for planar graphs with girth at least 5. We also show that chi'(alpha) (G) <= Delta (G) + 12 for all planar G, which improves a previous result by Fiedorowicz, Haluszczak, and Narayan Inform. Process. Lett., 108 (2008), pp. 412-417].

Boxicity of Line Graphs

The boxicity of a graph H, denoted by View the MathML source, is the minimum integer k such that H is an intersection graph of axis-parallel k-dimensional boxes in View the MathML source. In this paper we show that for a line graph G of a multigraph, View the MathML source, where Δ(G) denotes the maximum degree of G. Since G is a line graph, Δ(G)≤2(χ(G)−1), where χ(G) denotes the chromatic number of G, and therefore, View the MathML source. For the d-dimensional hypercube Qd, we prove that View the MathML source. The question of finding a nontrivial lower bound for View the MathML source was left open by Chandran and Sivadasan in [L. Sunil Chandran, Naveen Sivadasan, The cubicity of Hypercube Graphs. Discrete Mathematics 308 (23) (2008) 5795–5800]. The above results are consequences of bounds that we obtain for the boxicity of a fully subdivided graph (a graph that can be obtained by subdividing every edge of a graph exactly once).

Boxicity of line graphs

The boxicity of a graph H, denoted by box(H), is the minimum integer k such that H is an intersection graph of axis-parallel k-dimensional boxes in R(k). In this paper we show that for a line graph G of a multigraph, box(G) <= 2 Delta (G)(inverted right perpendicularlog(2) log(2) Delta(G)inverted left perpendicular + 3) + 1, where Delta(G) denotes the maximum degree of G. Since G is a line graph, Delta(G) <= 2(chi (G) - 1), where chi (G) denotes the chromatic number of G, and therefore, box(G) = 0(chi (G) log(2) log(2) (chi (G))). For the d-dimensional hypercube Q(d), we prove that box(Q(d)) >= 1/2 (inverted right perpendicularlog(2) log(2) dinverted left perpendicular + 1). The question of finding a nontrivial lower bound for box(Q(d)) was left open by Chandran and Sivadasan in [L. Sunil Chandran, Naveen Sivadasan, The cubicity of Hypercube Graphs. Discrete Mathematics 308 (23) (2008) 5795-5800]. The above results are consequences of bounds that we obtain for the boxicity of a fully subdivided graph (a graph that can be obtained by subdividing every edge of a graph exactly once). (C) 2011 Elsevier B.V. All rights reserved.

Boxicity and Poset Dimension

Let G be a simple, undirected, finite graph with vertex set V (G) and edge set E(G). A k-dimensional box is a Cartesian product of closed intervals [a(1), b(1)] x [a(2), b(2)] x ... x [a(k), b(k)]. The boxicity of G, box(G), is the minimum integer k such that G can be represented as the intersection graph of k-dimensional boxes; i.e., each vertex is mapped to a k-dimensional box and two vertices are adjacent in G if and only if their corresponding boxes intersect. Let P = (S, P) be a poset, where S is the ground set and P is a reflexive, antisymmetric and transitive binary relation on S. The dimension of P, dim(P), is the minimum integer t such that P can be expressed as the intersection of t total orders. Let G(P) be the underlying comparability graph of P; i.e., S is the vertex set and two vertices are adjacent if and only if they are comparable in P. It is a well-known fact that posets with the same underlying comparability graph have the same dimension. The first result of this paper links the dimension of a poset to the boxicity of its underlying comparability graph. In particular, we show that for any poset P, box(G(P))/(chi(G(P)) - 1) <= dim(P) <= 2box(G(P)), where chi(G(P)) is the chromatic number of G(P) and chi(G(P)) not equal 1. It immediately follows that if P is a height-2 poset, then box(G(P)) <= dim(P) <= 2box(G(P)) since the underlying comparability graph of a height-2 poset is a bipartite graph. The second result of the paper relates the boxicity of a graph G with a natural partial order associated with the extended double cover of G, denoted as G(c): Note that G(c) is a bipartite graph with partite sets A and B which are copies of V (G) such that, corresponding to every u is an element of V (G), there are two vertices u(A) is an element of A and u(B) is an element of B and {u(A), v(B)} is an edge in G(c) if and only if either u = v or u is adjacent to v in G. Let P(c) be the natural height-2 poset associated with G(c) by making A the set of minimal elements and B the set of maximal elements. We show that box(G)/2 <= dim(P(c)) <= 2box(G) + 4. These results have some immediate and significant consequences. The upper bound dim(P) <= 2box(G(P)) allows us to derive hitherto unknown upper bounds for poset dimension such as dim(P) = 2 tree width (G(P)) + 4, since boxicity of any graph is known to be at most its tree width + 2. In the other direction, using the already known bounds for partial order dimension we get the following: (1) The boxicity of any graph with maximum degree Delta is O(Delta log(2) Delta), which is an improvement over the best-known upper bound of Delta(2) + 2. (2) There exist graphs with boxicity Omega(Delta log Delta). This disproves a conjecture that the boxicity of a graph is O(Delta). (3) There exists no polynomial-time algorithm to approximate the boxicity of a bipartite graph on n vertices with a factor of O(n(0.5-is an element of)) for any is an element of > 0 unless NP = ZPP.

On k-stellated and k-stacked spheres

We introduce the class Sigma(k)(d) of k-stellated (combinatorial) spheres of dimension d (0 <= k <= d + 1) and compare and contrast it with the class S-k(d) (0 <= k <= d) of k-stacked homology d-spheres. We have E-1(d) = S-1(d), and Sigma(k)(d) subset of S-k(d) ford >= 2k-1. However, for each k >= 2 there are k-stacked spheres which are not k-stellated. For d <= 2k - 2, the existence of k-stellated spheres which are not k-stacked remains an open question. We also consider the class W-k(d) (and K-k(d)) of simplicial complexes all whose vertex-links belong to Sigma(k)(d - 1) (respectively, S-k(d - 1)). Thus, W-k(d) subset of K-k(d) for d >= 2k, while W-1(d) = K-1(d). Let (K) over bar (k)(d) denote the class of d-dimensional complexes all whose vertex-links are k-stacked balls. We show that for d >= 2k + 2, there is a natural bijection M -> (M) over bar from K-k(d) onto (K) over bar (k)(d + 1) which is the inverse to the boundary map partial derivative: (K) over bar (k)(d + 1) -> (K) over bar (k)(d). Finally, we complement the tightness results of our recent paper, Bagchi and Datta (2013) 5], by showing that, for any field F, an F-orientable (k + 1)-neighbourly member of W-k(2k + 1) is F-tight if and only if it is k-stacked.

Bipartite Powers of k-chordal Graphs

Let k be an integer and k >= 3. A graph G is k-chordal if G does not have an induced cycle of length greater than k. From the definition it is clear that 3-chordal graphs are precisely the class of chordal graphs. Duchet proved that, for every positive integer m, if G m is chordal then so is G(m+2). Brandst `` adt et al. in Andreas Brandsadt, Van Bang Le, and Thomas Szymczak. Duchet- type theorems for powers of HHD- free graphs. Discrete Mathematics, 177(1- 3): 9- 16, 1997.] showed that if G m is k - chordal, then so is G(m+2). Powering a bipartite graph does not preserve its bipartitedness. In order to preserve the bipartitedness of a bipartite graph while powering Chandran et al. introduced the notion of bipartite powering. This notion was introduced to aid their study of boxicity of chordal bipartite graphs. The m - th bipartite power G(m]) of a bipartite graph G is the bipartite graph obtained from G by adding edges (u; v) where d G (u; v) is odd and less than or equal to m. Note that G(m]) = G(m+1]) for each odd m. In this paper we show that, given a bipartite graph G, if G is k-chordal then so is G m], where k, m are positive integers with k >= 4

On Additive Combinatorics of Permutations of Z(n)

Let Z(n) denote the ring of integers modulo n. A permutation of Z(n) is a sequence of n distinct elements of Z(n). Addition and subtraction of two permutations is defined element-wise. In this paper we consider two extremal problems on permutations of Z(n), namely, the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is again a permutation, and the maximum size of a collection of permutations such that no sum of two distinct permutations in the collection is a permutation. Let the sizes be denoted by s (n) and t (n) respectively. The case when n is even is trivial in both the cases, with s (n) = 1 and t (n) = n!. For n odd, we prove (n phi(n))/2(k) <= s(n) <= n!.2(-)(n-1)/2/((n-1)/2)! and 2 (n-1)/2 . (n-1/2)! <= t (n) <= 2(k) . (n-1)!/phi(n), where k is the number of distinct prime divisors of n and phi is the Euler's totient function.

REPRESENTING A CUBIC GRAPH AS THE INTERSECTION GRAPH OF AXIS-PARALLEL BOXES IN THREE DIMENSIONS

We show that every graph of maximum degree 3 can be represented as the intersection graph of axis parallel boxes in three dimensions, that is, every vertex can be mapped to an axis parallel box such that two boxes intersect if and only if their corresponding vertices are adjacent. In fact, we construct a representation in which any two intersecting boxes touch just at their boundaries.

Rainbow matchings in strongly edge-colored graphs

A rainbow matching of an edge-colored graph G is a matching in which no two edges have the same color. There have been several studies regarding the maximum size of a rainbow matching in a properly edge-colored graph G in terms of its minimum degree 3(G). Wang (2011) asked whether there exists a function f such that a properly edge-colored graph G with at least f (delta(G)) vertices is guaranteed to contain a rainbow matching of size delta(G). This was answered in the affirmative later: the best currently known function Lo and Tan (2014) is f(k) = 4k - 4, for k >= 4 and f (k) = 4k - 3, for k <= 3. Afterwards, the research was focused on finding lower bounds for the size of maximum rainbow matchings in properly edge-colored graphs with fewer than 4 delta(G) - 4 vertices. Strong edge-coloring of a graph G is a restriction of proper edge-coloring where every color class is required to be an induced matching, instead of just being a matching. In this paper, we give lower bounds for the size of a maximum rainbow matching in a strongly edge-colored graph Gin terms of delta(G). We show that for a strongly edge-colored graph G, if |V(G)| >= 2 |3 delta(G)/4|, then G has a rainbow matching of size |3 delta(G)/4|, and if |V(G)| < 2 |3 delta(G)/4|, then G has a rainbow matching of size |V(G)|/2] In addition, we prove that if G is a strongly edge-colored graph that is triangle-free, then it contains a rainbow matching of size at least delta(G). (C) 2015 Elsevier B.V. All rights reserved.

SEPARATION DIMENSION OF BOUNDED DEGREE GRAPHS

The separation dimension of a graph G is the smallest natural number k for which the vertices of G can be embedded in R-k such that any pair of disjoint edges in G can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family F of total orders of the vertices of G such that for any two disjoint edges of G, there exists at least one total order in F in which all the vertices in one edge precede those in the other. In general, the maximum separation dimension of a graph on n vertices is Theta(log n). In this article, we focus on bounded degree graphs and show that the separation dimension of a graph with maximum degree d is at most 2(9) (log*d)d. We also demonstrate that the above bound is nearly tight by showing that, for every d, almost all d-regular graphs have separation dimension at least d/2]

Two player game variant of the Erdos-Szekeres problem

The classical Erdos-Szekeres theorem states that a convex k-gon exists in every sufficiently large point set. This problem has been well studied and finding tight asymptotic bounds is considered a challenging open problem. Several variants of the Erdos-Szekeres problem have been posed and studied in the last two decades. The well studied variants include the empty convex k-gon problem, convex k-gon with specified number of interior points and the chromatic variant. In this paper, we introduce the following two player game variant of the Erdos-Szekeres problem: Consider a two player game where each player playing in alternate turns, place points in the plane. The objective of the game is to avoid the formation of the convex k-gon among the placed points. The game ends when a convex k-gon is formed and the player who placed the last point loses the game. In our paper we show a winning strategy for the player who plays second in the convex 5-gon game and the empty convex 5-gon game by considering convex layer configurations at each step. We prove that the game always ends in the 9th step by showing that the game reaches a specific set of configurations.

DETERMINISTIC ALGORITHMS FOR MATCHING AND PACKING PROBLEMS BASED ON REPRESENTATIVE SETS

In this work, we study the well-known r-DIMENSIONAL k-MATCHING ((r, k)-DM), and r-SET k-PACKING ((r, k)-SP) problems. Given a universe U := U-1 ... U-r and an r-uniform family F subset of U-1 x ... x U-r, the (r, k)-DM problem asks if F admits a collection of k mutually disjoint sets. Given a universe U and an r-uniform family F subset of 2(U), the (r, k)-SP problem asks if F admits a collection of k mutually disjoint sets. We employ techniques based on dynamic programming and representative families. This leads to a deterministic algorithm with running time O(2.851((r-1)k) .vertical bar F vertical bar. n log(2)n . logW) for the weighted version of (r, k)-DM, where W is the maximum weight in the input, and a deterministic algorithm with running time O(2.851((r-0.5501)k).vertical bar F vertical bar.n log(2) n . logW) for the weighted version of (r, k)-SP. Thus, we significantly improve the previous best known deterministic running times for (r, k)-DM and (r, k)-SP and the previous best known running times for their weighted versions. We rely on structural properties of (r, k)-DM and (r, k)-SP to develop algorithms that are faster than those that can be obtained by a standard use of representative sets. Incorporating the principles of iterative expansion, we obtain a better algorithm for (3, k)-DM, running in time O(2.004(3k).vertical bar F vertical bar . n log(2)n). We believe that this algorithm demonstrates an interesting application of representative families in conjunction with more traditional techniques. Furthermore, we present kernels of size O(e(r)r(k-1)(r) logW) for the weighted versions of (r, k)-DM and (r, k)-SP, improving the previous best known kernels of size O(r!r(k-1)(r) logW) for these problems.

Upper bound on cubicity in terms of boxicity for graphs of low chromatic number

The boxicity (respectively cubicity) of a graph G is the least integer k such that G can be represented as an intersection graph of axis-parallel k-dimensional boxes (respectively k-dimensional unit cubes) and is denoted by box(G) (respectively cub(G)). It was shown by Adiga and Chandran (2010) that for any graph G, cub(G) <= box(G) log(2) alpha(G], where alpha(G) is the maximum size of an independent set in G. In this note we show that cub(G) <= 2 log(2) X (G)] box(G) + X (G) log(2) alpha(G)], where x (G) is the chromatic number of G. This result can provide a much better upper bound than that of Adiga and Chandran for graph classes with bounded chromatic number. For example, for bipartite graphs we obtain cub(G) <= 2(box(G) + log(2) alpha(G)] Moreover, we show that for every positive integer k, there exist graphs with chromatic number k such that for every epsilon > 0, the value given by our upper bound is at most (1 + epsilon) times their cubicity. Thus, our upper bound is almost tight. (c) 2015 Elsevier B.V. All rights reserved.