Achromatic number of fragmentable graphs

A complete coloring of a simple graph G is a proper vertex coloring such that each pair of colors appears together on at least one edge. The achromatic number ψ(G) is the greatest number of colors in such a coloring. We say a class of graphs is fragmentable if for any positive ε, there is a constant C such that any graph in the class can be broken into pieces of size at most C by removing a proportion at most ε of the vertices. Examples include planar graphs and grids of fixed dimension. Determining the achromatic number of a graph is NP‐complete in general, even for trees, and the achromatic number is known precisely for only very restricted classes of graphs. We extend these classes very considerably, by giving, for graphs in any class which is fragmentable, triangle‐free, and of bounded degree, a necessary and sufficient condition for a sufficiently large graph to have a complete coloring with a given number of colors. For the same classes, this gives a tight lower bound for the achromatic number of sufficiently large graphs, and shows that the achromatic number can be determined in polynomial time. As examples, we give exact values of the achromatic number for several graph families. © 2009 Wiley Periodicals, Inc. J Graph Theory 65:94–114, 2010     

Graph with given achromatic number

An irreducible (point-determining) graph is one in which distinct vertices have distinct neighbourhoods. Every graph X can be reduced to an irreducible graph X¹ by identifying all vertices with the same neighbourhood; the colourability properties of X¹ carry over to X. Hence irreducible graphs are instrumental in the study of achromatic number. We prove that there are only finitely many irreducible graphs with a given achromatic number, and describe all graphs with achromatic number less than four. We deduce certain bounds on the achromatic number X in terms of the number of vertices of X¹. In the course of the proofs we calculate the achromatic numbers of paths and cycles. Generalizations of the main theorem to homomorphisms other than colourings are discussed.     

The chromatic number and other functions of the lexicographic product

The values of the chromatic and achromatic number, point- and line-connectivity, and point independence number of the lexicographic product of two graphs are examined in relation to the values of the respective parameters on the factor graphs.     

Geometric Achromatic and Pseudoachromatic Indices

The pseudoachromatic index of a graph is the maximum number of colors that can be assigned to its edges, such that each pair of different colors is incident to a common vertex. If for each vertex its incident edges have different color, then this maximum is known as achromatic index. Both indices have been widely studied. A geometric graph is a graph drawn in the plane such that its vertices are points in general position, and its edges are straight-line segments. In this paper we extend the notion of pseudoachromatic and achromatic indices for geometric graphs, and present results for complete geometric graphs. In particular, we show that for n points in convex position the achromatic index and the pseudoachromatic index of the complete geometric graph are \(\lfloor \frac{n^2+n}{4} \rfloor \).     

Difference systems,graph designs,and coloring problems

Difference systems and chromatic difference systems are generalizations of difference sets. They may be used to construct various types of graphical block designs and to determine graphs with maximal achromatic number. The bulk of the paper is concerned with the application of these ideas to certain special types of graphs: forests, paths, circuits, and generalized Petersen graphs. Cyclotomy is an effective tool in the construction of difference systems for graphs with a high degree of symmetry.     

Some perfect coloring properties of graphs

A Grundy n-coloring of a finite graph is a coloring of the points of the graph with the non-negative integers smaller than n such that each point is adjacent to some point of each smaller color but to none of the same color. The Grundy number of a graph is the maximum n for which it has a Grundy n-coloring. Characterizations are given of the families of finite graphs G such that for each induced subgraph H of G: (1) the Grundy number of H is equal to the chromatic number of H; (2) the Grundy number of H is equal to the maximum clique size of H; (3) the achromatic number of H is equal to the chromatic number of H; (4) the achromatic number of H is equal to the maximum clique size of H. The definitions are further extended to infinite graphs, and some of the above characterizations are shown to be true for denumerable graphs and locally finite graphs.     

Polar permutation graphs are polynomial-time recognisable

Polar graphs generalise bipartite graphs, cobipartite graphs, and split graphs, and they constitute a special type of matrix partitions. A graph is polar if its vertex set can be partitioned into two, such that one part induces a complete multipartite graph and the other part induces a disjoint union of complete graphs. Deciding whether a given arbitrary graph is polar, is an NP$\text{NP}$-complete problem. Here, we show that for permutation graphs this problem can be solved in polynomial time. The result is surprising, as related problems like achromatic number and cochromatic number are NP$\text{NP}$-complete on permutation graphs. We give a polynomial-time algorithm for recognising graphs that are both permutation and polar. Prior to our result, polarity has been resolved only for chordal graphs and cographs.     

On the Achromatic Index of for a Prime <Emphasis Type="Italic">q</Emphasis>

The achromatic index of a graph G is the maximum number of colours that can be assigned to edges of G in such a way that the colouring is proper and any two colours meet on two adjacent edges. To compute the achromatic index of complete graphs seems to be a very difficult task. Indeed, that knowledge would yield all odd projective plane orders. The aim of the paper is to establish a nontrivial lower bound of the achromatic index of for a prime q7.Acknowledgments The first author gratefully acknowledges a support of the Slovak grant VEGA 1/7467/20. The authors wish to thank to an anonymous referee for an easy proof of Lemma 8.Final version received: November 10, 2003     

On Sturmian graphs

In this paper we define Sturmian graphs and we prove that all of them have a certain “counting” property. We show deep connections between this counting property and two conjectures, by Moser and by Zaremba, on the continued fraction expansion of real numbers. These graphs turn out to be the underlying graphs of compact directed acyclic word graphs of central Sturmian words. In order to prove this result, we give a characterization of the maximal repeats of central Sturmian words. We show also that, in analogy with the case of Sturmian words, these graphs converge to infinite ones.     

The Index of a Periodic Graph

For periodic graphs, a special class of infinite, but locally finite graphs, an index theory is developed that can serve in classifying these graphs and that enables connections with various graph invariants as in the case of finite graphs. The index is defined with the aid of certain finite matrices that result rather canonically from reductions of the infinite adjacency operator due to the periodicity. As a central result we derive a sharp global lower bound for the index of any periodic graph.     

Partitioning Perfect Graphs into Stars

The partition of graphs into “nice” subgraphs is a central algorithmic problem with strong ties to matching theory. We study the partitioning of undirected graphs into same‐size stars, a problem known to be NP‐complete even for the case of stars on three vertices. We perform a thorough computational complexity study of the problem on subclasses of perfect graphs and identify several polynomial‐time solvable cases, for example, on interval graphs and bipartite permutation graphs, and also NP‐complete cases, for example, on grid graphs and chordal graphs.     

Existence of proportional graphs

The notion of p-proportional graphs comes from the study of subgraph counts in random graphs, where the p-proportional graphs occur as exceptional cases in a central limit theorem. In this paper we show that p-proportional graphs exist for all rational p, 0 < p < 1, and thereby proving a conjecture from [3]. © 1993 John Wiley & Sons, Inc.     

Polarity of chordal graphs

Polar graphs are a common generalization of bipartite, cobipartite, and split graphs. They are defined by the existence of a certain partition of vertices, which is NP-complete to decide for general graphs. It has been recently proved that for cographs, the existence of such a partition can be characterized by finitely many forbidden subgraphs, and hence tested in polynomial time. In this paper we address the question of polarity of chordal graphs, arguing that this is in essence a question of colourability, and hence chordal graphs are a natural restriction. We observe that there is no finite forbidden subgraph characterization of polarity in chordal graphs; nevertheless we present a polynomial time algorithm for polarity of chordal graphs. We focus on a special case of polarity (called monopolarity) which turns out to be the central concept for our algorithms. For the case of monopolar graphs, we illustrate the structure of all minimal obstructions; it turns out that they can all be described by a certain graph grammar, permitting our monopolarity algorithm to be cast as a certifying algorithm.     

Almost self-centered graphs

The center of a graph is the set of vertices with minimum eccentricity. Graphs in which all vertices are central are called self-centered graphs. In this paper almost self-centered (ASC) graphs are introduced as the graphs with exactly two non-central vertices. The block structure of these graphs is described and constructions for generating such graphs are proposed. Embeddings of arbitrary graphs into ASC graphs are studied. In particular it is shown that any graph can be embedded into an ASC graph of prescribed radius. Embeddings into ASC graphs of radius two are studied in more detail. ASC index of a graph G is introduced as the smallest number of vertices needed to add to G such that G is an induced subgraph of an ASC graph.     

On the Congruences of Some Combinatorial Numbers

In this paper, we apply Lucas' theorem to evaluate the congruences of several combinatorial numbers, including the central Delannoy numbers and a class of Apéry-like numbers, the numbers of noncrossing connected graphs, the numbers of total edges of all noncrossing connected graphs on n vertices, etc. One of these results verifies a conjecture given by Deutsch and Sagan recently. In the end, we use an automaton to explain the idea of our approach.     

Polar graphs and maximal independent sets

We study two central problems of algorithmic graph theory: finding maximum and minimum maximal independent sets. Both problems are known to be NP-hard in general. Moreover, they remain NP-hard in many special classes of graphs. For instance, the problem of finding minimum maximal independent sets has been recently proven to be NP-hard in the class of so-called (1,2)-polar graphs. On the other hand, both problems can be solved in polynomial time for (1,1)-polar, also known as split graphs. In this paper, we address the question of distinguishing new classes of graphs admitting polynomial-time solutions for the two problems in question. To this end, we extend the hierarchy of (α,β)-polar graphs and study the computational complexity of the problems on polar graphs of special types.     

Approximation Algorithms for the Achromatic Number

The achromatic number for a graph G = V, E is the largest integer m such that there is a partition of V into disjoint independent sets {V₁, …, V_m} such that for each pair of distinct sets V_i, V_j, V_i V_j is not an independent set in G. Yannakakis and Gavril (1980, SIAM J. Appl. Math.38, 364–372) proved that determining this value for general graphs is NP-complete. For n-vertex graphs we present the first o(n) approximation algorithm for this problem. We also present an O(n^5/12) approximation algorithm for graphs with girth at least 5 and a constant approximation algorithm for trees.     

Computation of a (canonical) doubly perfect elimination ordering of a doubly chordal graph

The class ofdoubly chordal graphs is a subclass ofchordal graphs and a superclass ofstrongly chordal graphs, which arise in so many application areas. Many optimization problems like domination and Steiner tree are NP-complete on chordal graphs but can be solved in polynomial time on doubly chordal graphs. The central to designing efficient algorithms for doulby chordal graphs is the concept of(canonical) doubly perfect elimination orderings. We present linear time algorithms to compute a(canonical) doubly perfect elimination ordering of adoubly chordal graph.     

Vizing's conjecture: a survey and recent results

Vizing's conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this paper we survey the approaches to this central conjecture from domination theory and give some new results along the way. For instance, several new properties of a minimal counterexample to the conjecture are obtained and a lower bound for the domination number is proved for products of claw‐free graphs with arbitrary graphs. Open problems, questions and related conjectures are discussed throughout the paper. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 46–76, 2012     

Minimally non-Pfaffian graphs

We consider the question of characterizing Pfaffian graphs. We exhibit an infinite family of non-Pfaffian graphs minimal with respect to the matching minor relation. This is in sharp contrast with the bipartite case, as Little [C.H.C. Little, A characterization of convertible (0,1)-matrices, J. Combin. Theory Ser. B 18 (1975) 187–208] proved that every bipartite non-Pfaffian graph contains a matching minor isomorphic to K_3,3. We relax the notion of a matching minor and conjecture that there are only finitely many (perhaps as few as two) non-Pfaffian graphs minimal with respect to this notion.We define Pfaffian factor-critical graphs and study them in the second part of the paper. They seem to be of interest as the number of near perfect matchings in a Pfaffian factor-critical graph can be computed in polynomial time. We give a polynomial time recognition algorithm for this class of graphs and characterize non-Pfaffian factor-critical graphs in terms of forbidden central subgraphs.