Acyclic chromatic index of planar graphs with triangles

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by , is the least number of colors in an acyclic edge coloring of G. Let G be a planar graph with maximum degree Δ(G). In this paper, we show that , if G contains no 4-cycle; , if G contains no intersecting triangles; and if G contains no adjacent triangles.     

Weighted Coloring: further complexity and approximability results

Given a vertex-weighted graph G=(V,E;w), w(v)?0 for any v∈V, we consider a weighted version of the coloring problem which consists in finding a partition S=(S₁,…,Sk) of the vertex set of G into stable sets and minimizing where the weight of S is defined as . In this paper, we continue the investigation of the complexity and the approximability of this problem by answering some of the questions raised by Guan and Zhu [D.J. Guan, X. Zhu, A coloring problem for weighted graphs, Inform. Process. Lett. 61 (2) (1997) 77-81].     

Note on the connectivity of line graphs

Let G be a connected graph with vertex set V(G), edge set E(G), vertex-connectivity κ(G) and edge-connectivity λ(G).A subset S of E(G) is called a restricted edge-cut if G−S is disconnected and each component contains at least two vertices. The restricted edge-connectivity λ₂(G) is the minimum cardinality over all restricted edge-cuts. Clearly λ₂(G)?λ(G)?κ(G).In 1969, Chartrand and Stewart have shown that

     

About acyclic edge colourings of planar graphs

Let G=(V,E) be any finite simple graph. A mapping is called an acyclic edge k-colouring of G, if any two adjacent edges have different colours and there are no bichromatic cycles in G. In other words, for every pair of distinct colours i and j, the subgraph induced by all the edges which have either colour i or j is acyclic. The smallest number k of colours, such that G has an acyclic edge k-colouring is called the acyclic chromatic index of G and is denoted by .In 1991, Alon et al. [N. Alon, C.J.H. McDiarmid, B.A. Reed, Acyclic coloring of graphs, Random Structures and Algorithms 2 (1991) 277-288] proved that for any graph G of maximum degree Δ(G). This bound was later improved to 16Δ(G) by Molloy and Reed in [M. Molloy, B. Reed, Further algorithmic aspects of the local lemma, in: Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 1998, pp. 524-529].In this paper we prove that for a planar graph G without cycles of length three and that the same holds if G has an edge-partition into two forests. We also show that if G is planar.     

On edge covering colorings of graphs

An edge covering coloring of a graph G is an edge-coloring of G such that each color appears at each vertex at least one time. The maximum integer k such that G has an edge covering coloring with k colors is called the edge covering chromatic index of G and denoted by . It is known that for any graph G with minimum degree δ(G), it holds that . Based on the subgraph of G induced by the vertices of minimum degree, we find a new sufficient condition for a graph G to satisfy . This result substantially extends a result of Wang et al. in 2006.     

Implicit B-trees: a new data structure for the dictionary problem

An implicit data structure for the dictionary problem maintains n data values in the first n locations of an array in such a way that it efficiently supports the operations insert, delete and search. No information other than that in O(1) memory cells and in the input data is to be retained; and the only operations performed on the data values (other than reads and writes) are comparisons. This paper describes the implicit B-tree, a new data structure supporting these operations in block transfers like in regular B-trees, under the realistic assumption that a block stores keys, so that reporting r consecutive keys in sorted order has a cost of block transfers. En route a number of space efficient techniques for handling segments of a large array in a memory hierarchy are developed. Being implicit, the proposed data structure occupies exactly ⌈n/B⌉ blocks of memory after each update, where n is the number of keys after each update and B is the number of keys contained in a memory block. In main memory, the time complexity of the operations is , disproving a conjecture of the mid 1980s.     

Acyclic edge colourings of graphs with the number of edges linearly bounded by the number of vertices

Let G be any finite graph. A mapping c:E(G)→{1,…,k} is called an acyclic edge k-colouring of G, if any two adjacent edges have different colours and there are no bichromatic cycles in G. In other words, for every pair of distinct colours i and j, the subgraph induced in G by all the edges that have colour i or j is acyclic. The smallest number k of colours such that G has an acyclic edge k-colouring is called the acyclic chromatic index of G and is denoted by .Determining the acyclic chromatic index of a graph is a hard problem, both from theoretical and algorithmical point of view. In 1991, Alon et al. proved that for any graph G of maximum degree Δ(G). This bound was later improved to 16Δ(G) by Molloy and Reed. In general, the problem of computing the acyclic chromatic index of a graph is NP-complete. Only a few algorithms for finding acyclic edge colourings have been known by now. Moreover, these algorithms work only for graphs from particular classes.In our paper, we prove that for every graph G which satisfies the condition that |E(G^′)|?t|V(G^′)|−1 for each subgraph G^′⊆G, where t?2 is a given integer, the constant p=2t³−3t+2. Based on that result, we obtain a polynomial algorithm which computes such a colouring. The class of graphs covered by our theorem is quite rich, for example, it contains all t-degenerate graphs.     

Feedback vertex sets in star graphs

In a graph G=(V,E), a subset F⊂V(G) is a feedback vertex set of G if the subgraph induced by V(G)?F is acyclic. In this paper, we propose an algorithm for finding a small feedback vertex set of a star graph. Indeed, our algorithm can derive an upper bound to the size of the feedback vertex set for star graphs. Also by applying the properties of regular graphs, a lower bound can easily be achieved for star graphs.     

Quasi-random rumor spreading: Reducing randomness can be costly

We give a time-randomness tradeoff for the quasi-random rumor spreading protocol proposed by Doerr, Friedrich and Sauerwald [SODA 2008] on complete graphs. In this protocol, the goal is to spread a piece of information originating from one vertex throughout the network. Each vertex is assumed to have a (cyclic) list of its neighbors. Once a vertex is informed by one of its neighbors, it chooses a position in its list uniformly at random and then informs its neighbors starting from that position and proceeding in order of the list. Angelopoulos, Doerr, Huber and Panagiotou [Electron. J. Combin. 2009] showed that after rounds, the rumor will have been broadcasted to all nodes with probability 1−o(1).We study the broadcast time when the amount of randomness available at each node is reduced in natural way. In particular, we prove that if each node can only make its initial random selection from every ?-th node on its list, then there exists lists such that steps are needed to inform every vertex with probability at least . This shows that a further reduction of the amount of randomness used in a simple quasi-random protocol comes at a loss of efficiency.     

An approximation algorithm dependent on edge-coloring number for minimum maximal matching problem

We consider the minimum maximal matching problem, which is NP-hard (Yannakakis and Gavril (1980) [18]). Given an unweighted simple graph G=(V,E), the problem seeks to find a maximal matching of minimum cardinality. It was unknown whether there exists a non-trivial approximation algorithm whose approximation ratio is less than 2 for any simple graph. Recently, Z. Gotthilf et al. (2008) [5] presented a -approximation algorithm, where c is an arbitrary constant.In this paper, we present a -approximation algorithm based on an LP relaxation, where χ^′(G) is the edge-coloring number of G. Our algorithm is the first non-trivial approximation algorithm whose approximation ratio is independent of |V|. Moreover, it is known that the minimum maximal matching problem is equivalent to the edge dominating set problem. Therefore, the edge dominating set problem is also -approximable. From edge-coloring theory, the approximation ratio of our algorithm is , where Δ(G) represents the maximum degree of G. In our algorithm, an LP formulation for the edge dominating set problem is used. Fujito and Nagamochi (2002) [4] showed the integrality gap of the LP formulation for bipartite graphs is at least . Moreover, χ^′(G) is Δ(G) for bipartite graphs. Thus, as far as an approximation algorithm for the minimum maximal matching problem uses the LP formulation, we believe our result is the best possible.     

Signed and minus clique-transversal functions on graphs

A minus (respectively, signed) clique-transversal function of a graph G=(V,E) is a function (respectively, {−1,1}) such that _∑u∈Cf(u)?1 for every maximal clique C of G. The weight of a minus (respectively, signed) clique-transversal function of G is f(V)=_∑v∈Vf(v). The minus (respectively, signed) clique-transversal problem is to find a minus (respectively, signed) clique-transversal function of G of minimum weight. In this paper, we present a unified approach to these two problems on strongly chordal graphs. Notice that trees, block graphs, interval graphs, and directed path graphs are subclasses of strongly chordal graphs. We also prove that the signed clique-transversal problem is NP-complete for chordal graphs and planar graphs.     

Enhanced algorithms for Local Search

Let G=(V,E) be a finite graph, and be any function. The Local Search problem consists in finding a local minimum of the function f on G, that is a vertex v such that f(v) is not larger than the value of f on the neighbors of v in G. In this note, we first prove a separation theorem slightly stronger than the one of Gilbert, Hutchinson and Tarjan for graphs of constant genus. This result allows us to enhance a previously known deterministic algorithm for Local Search with query complexity , so that we obtain a deterministic query complexity of , where n is the size of G, d is its maximum degree, and g is its genus. We also give a quantum version of our algorithm, whose query complexity is of . Our deterministic and quantum algorithms have query complexities respectively smaller than the algorithm Randomized Steepest Descent of Aldous and Quantum Steepest Descent of Aaronson for large classes of graphs, including graphs of bounded genus and planar graphs.     

A greedy approach to compute a minimum cycle basis of a directed graph

We consider the problem of computing a minimum cycle basis of a directed graph with m arcs and n nodes. We adapt the greedy approach proposed by Horton [A polynomial-time algorithm to find the shortest cycle basis of a graph, SIAM J. Comput. 16 (1987) 358] and hereby obtain a very simple exact algorithm of complexity , being as fast as the first algorithm proposed for this problem [A polynomial time algorithm for minimum cycle basis in directed graphs, Kurt Mehlhorn's List of Publications, 185, MPI, Saarbrücken, 2004, http://www.mpi-sb.mpg.de/~mehlhorn/ftp/DirCycleBasis.ps; Proc. STACS 2005, submitted for publication]. Moreover, the speed-up of Golynski and Horton [A polynomial time algorithm to find the minimum cycle basis of a regular matroid, in: M. Penttonen, E. Meineche Schmidt (Eds.), SWAT 2002, Lecture Notes in Comput. Sci., vol. 2368, Springer, Berlin, 2002, pp. 200-209] applies to this problem, providing an exact algorithm of complexity , in particular . Finally, we prove that these greedy approaches fail for more specialized subclasses of directed cycle bases.     

Ranking numbers of graphs

Given a graph G, a vertex ranking (or simply, ranking) of G is a mapping f from V(G) to the set of all positive integers, such that for any path between two distinct vertices u and v with f(u)=f(v), there is a vertex w in the path with f(w)>f(u). If f is a ranking of G, the ranking number of G under f, denoted γf(G), is defined by , and the ranking number of G, denoted γ(G), is defined by . The vertex ranking problem is to determine the ranking number γ(G) of a given graph G. This problem is a natural model for the manufacturing scheduling problem. We study the ranking numbers of graphs in this paper. We consider the relation between the ranking numbers and the minimal cut sets, and the relation between the ranking numbers and the independent sets. From this, we obtain the ranking numbers of the powers of paths and the powers of cycles, the Cartesian product of P₂ with Pn or Cn, and the caterpilars. And we also find the vertex ranking numbers of the composition of two graphs in this paper.     

The labeled perfect matching in bipartite graphs

In this paper, we deal with both the complexity and the approximability of the labeled perfect matching problem in bipartite graphs. Given a simple graph G=(V,E) with |V|=2n vertices such that E contains a perfect matching (of size n), together with a color (or label) function , the labeled perfect matching problem consists in finding a perfect matching on G that uses a minimum or a maximum number of colors.