The clique operator on circular-arc graphs

A circular-arc graphG is the intersection graph of a collection of arcs on the circle and such a collection is called a model of G. Say that the model is proper when no arc of the collection contains another one, it is Helly when the arcs satisfy the Helly Property, while the model is proper Helly when it is simultaneously proper and Helly. A graph admitting a Helly (resp. proper Helly) model is called a Helly (resp. proper Helly) circular-arc graph. The clique graphK(G) of a graph G is the intersection graph of its cliques. The iterated clique graphKⁱ(G) of G is defined by K⁰(G)=G and Kⁱ⁺¹(G)=K(Kⁱ(G)). In this paper, we consider two problems on clique graphs of circular-arc graphs. The first is to characterize clique graphs of Helly circular-arc graphs and proper Helly circular-arc graphs. The second is to characterize the graph to which a general circular-arc graph K-converges, if it is K-convergent. We propose complete solutions to both problems, extending the partial results known so far. The methods lead to linear time recognition algorithms, for both problems.     

Linear upper bounds for local Ramsey numbers

A coloring of the edges of a graph is called alocal k-coloring if every vertex is incident to edges of at mostk distinct colors. For a given graphG, thelocal Ramsey number, r _loc ^k (G), is the smallest integern such that any localk-coloring ofK _n , (the complete graph onn vertices), contains a monochromatic copy ofG. The following conjecture of Gyárfás et al. is proved here: for each positive integerk there exists a constantc = c(k) such thatr _loc ^k (G) cr ^k (G), for every connected grraphG (wherer ^k (G) is theusual Ramsey number fork colors). Possible generalizations for hypergraphs are considered.On leave from the Institute of Mathematics, Technical University of Warsaw, Poland.While on leave at University of Louisville, Fall 1985.     

Restricted greedy clique decompositions and greedy clique decompositions ofK 4-free graphs

A greedy clique decomposition of a graph is obtained by removing maximal cliques from a graph one by one until the graph is empty. It has recently been shown that any greedy clique decomposition of a graph of ordern has at mostn ²/4 cliques. In this paper, we extend this result by showing that for any positive integerp, 3≤p any clique decomposisitioof a graph of ordern obtained by removing maximal cliques of order at leastp one by one until none remain, in which case the remaining edges are removed one by one, has at mostt _p-1( ⁿ ) cliques. Heret _p-1( ⁿ ) is the number of edges in the Turán graph of ordern, which has no complete subgraphs of orderp. In connection with greedy clique decompositions, P. Winkler conjectured that for any greedy clique decompositionC of a graphG of ordern the sum over the number of vertices in each clique ofC is at mostn ²/2. We prove this conjecture forK ₄-free graphs and show that in the case of equality forC andG there are only two possibilities:

1. G?K _n/2,n/2
2. G is complete 3-partite, where each part hasn/3 vertices.

We show that in either caseC is completely determined.     

Mortality of iterated Gallai graphs

For a finite or infinite graphG, theGallai graph (G) ofG is defined as the graph whose vertex set is the edge setE(G) ofG; two distinct edges ofG are adjacent in (G) if they are incident but do not span a triangle inG. For any positive integert, thetth iterated Gallai graph ^t (G) ofG is defined by ( ^t–1(G)), where ⁰(G):=G. A graph is said to beGallai-mortal if some of its iterated Gallai graphs finally equals the empty graph. In this paper we characterize Gallai-mortal graphs in several ways.     

Graphs isomorphic to subgraphs of their iterated line graphs

A graphG is said to be embeddable into a graphH, if there is an isomorphism ofG into a subgraph ofH. It is shown in this paper that every unicycle or tree which is neither a path norK _1,3 embeds in itsn-th iterated line graph forn1 or 2, 3, and that every other connected graph that embeds in itsn-th iterated line graph may be constructed from such an embedded unicycle or tree in a natural way. A special kind of embedding of graph into itsn-th iterated line graph, called incidence embedding, is studied. Moreover, it is shown that for every positive integerk, there exists a graphG such that (G) = , where (G) is the leastn1 for whichG embeds inL ⁿ(G).     

Diameters of iterated clique graphs of chordal graphs

The clique graph K(G) of a graph is the intersection graph of maximal cliques of G. The iterated clique graph Kⁿ(G) is inductively defined as K(K^n?1(G)) and K¹(G) = K(G). Let the diameter diam(G) be the greatest distance between all pairs of vertices of G. We show that diam(Kⁿ(G)) = diam(G) — n if G is a connected chordal graph and n ≤ diam(G). This generalizes a similar result for time graphs by Bruce Hedman.     

Clique graphs of time graphs

A simple, finite graph G is called a time graph (equivalently, an indifference graph) if there is an injective real function f on the vertices v(G) such that v_iv_j ∈ e(G) for v_i ≠ v_j if and only if |f(v_i) ? f(v_j)| ≤ 1. A clique of a graph G is a maximal complete subgraph of G. The clique graph K(G) of a graph G is the intersection graph of the cliques of G. It will be shown that the clique graph of a time graph is a time graph, and that every time graph is the clique graph of some time graph. Denote the clique graph of a clique graph of G by K²(G), and inductively, denote K(K^m?1(G)) by K^m(G). Define the index indx(G) of a connected time graph G as the smallest integer n such that Kⁿ(G) is the trivial graph. It will be shown that the index of a time graph is equal to its diameter. Finally, bounds on the diameter of a time graph will be derived.     

Self-clique Helly circular-arc graphs

A clique in a graph is a complete subgraph maximal under inclusion. The clique graph of a graph is the intersection graph of its cliques. A graph is self-clique when it is isomorphic to its clique graph. A circular-arc graph is the intersection graph of a family of arcs of a circle. A Helly circular-arc graph is a circular-arc graph admitting a model whose arcs satisfy the Helly property. In this note, we describe all the self-clique Helly circular-arc graphs.     

Isoperimetric numbers of graph bundles

The main aim of this paper is to give some upper and lower bounds for the isoperimetric numbers of graph coverings or graph bundles, with exact values in some special cases. In addition, we show that the isoperimetric number of any covering graph is not greater than that of the base graph. Mohar's theorem for the isoperimetric number of the cartesian product of a graph and a complete graph can be extended to a more general case: The isoperimetric numberi(G × K _2n) of the cartesian product of any graphG and a complete graphK _2n on even vertices is the minimum of the isoperimetric numberi(G) andn, and it is also a sharp lower bound of the isoperimetric numbers of all graph bundles over the graphG with fiberK _2n. Furthermore, ifn 2i(G) then the isoperimetric number of any graph bundle overG with fibreK _n is equal to the isoperimetric numberi(G) ofG. Partially supported by The Ministry of Education, Korea.     

Thep-intersection number of a complete bipartite graph and orthogonal double coverings of a clique

Thep-intersection graph of a collection of finite sets {S _i } _i=1 ⁿ is the graph with vertices 1, ...,n such thati, j are adjacent if and only if |S _i S _j |p. Thep-intersection number of a graphG, herein denoted _p (G), is the minimum size of a setU such thatG is thep-intersection graph of subsets ofU. IfG is the complete bipartite graphK _n,n andp2, then _p (K _{n, n} )(n ²+(2p–1)n)/p. Whenp=2, equality holds if and only ifK _n has anorthogonal double covering, which is a collection ofn subgraphs ofK _n , each withn–1 edges and maximum degree 2, such that each pair of subgraphs shares exactly one edge. By construction,K _n has a simple explicit orthogonal double covering whenn is congruent modulo 12 to one of {1, 2, 5, 7, 10, 11}.Research supported in part by ONR Grant N00014-5K0570.     

On the maximal number of certain subgraphs inK r -free graphs

Given two graphsH andG, letH(G) denote the number of subgraphs ofG isomorphic toH. We prove that ifH is a bipartite graph with a one-factor, then for every triangle-free graphG withn verticesH(G) H(T ₂(n)), whereT ₂(n) denotes the complete bipartite graph ofn vertices whose colour classes are as equal as possible. We also prove that ifK is a completet-partite graph ofm vertices,r > t, n max(m, r – 1), then there exists a complete (r – 1)-partite graphG* withn vertices such thatK(G) K(G*) holds for everyK _r -free graphG withn vertices. In particular, in the class of allK _r -free graphs withn vertices the complete balanced (r – 1)-partite graphT _r–1(n) has the largest number of subgraphs isomorphic toK _t (t < r),C ₄,K _2,3. These generalize some theorems of Turán, Erdös and Sauer.Dedicated to Paul Turán on his 80th Birthday     

Centers of chordal graphs

In a graphG = (V, E), theeccentricity e(S) of a subset S ismax _{x V} min _{y S} d(x, y); ande(x) stands fore({x}). Thediameter ofG ismax _{x V} e(x), theradius r(G) ofG ismin _{x V} e(x) and theclique radius cr(G) ismin e(K) whereK runs over all cliques. Thecenter ofG is the subgraph induced byC(G), the set of all verticesx withe(x) = r(G). Aclique center is a cliqueK withe(K) = cr(G). In this paper, we study the problem of determining the centers of chordal graphs. It is shown that the center of a connected chordal graph is distance invariant, biconnected and of diameter no more than 5. We also prove that2cr(G) d(G) 2cr(G) + 1 for any connected chordal graphG. This result implies a characterization of a biconnected chordal graph of diameter 2 and radius 1 to be the center of some chordal graph.Supported by the National Science Council of the Republic of China under grant NSC77-0208-M008-05     

Facets of the clique partitioning polytope

A subsetA of the edge set of a graphG = (V, E) is called a clique partitioning ofG is there is a partition of the node setV into disjoint setsW ¹,,W ^k such that eachW _i induces a clique, i.e., a complete (but not necessarily maximal) subgraph ofG, and such thatA = _i=1 ^k 1{uv|u, v W _i ,u v}. Given weightsw ^e for alle E, the clique partitioning problem is to find a clique partitioningA ofG such that _eA w _e is as small as possible. This problem—known to be-hard, see Wakabayashi (1986)—comes up, for instance, in data analysis, and here, the underlying graphG is typically a complete graph. In this paper we study the clique partitioning polytope of the complete graphK _n , i.e., is the convex hull of the incidence vectors of the clique partitionings ofK _n . We show that triangles, 2-chorded odd cycles, 2-chorded even wheels and other subgraphs ofK _n induce facets of. The theoretical results described here have been used to design an (empirically) efficient cutting plane algorithm with which large (real-world) instances of the clique partitioning problem could be solved. These computational results can be found in Grötschel and Wakabayashi (1989).     

On thel-connectivity of a graph

For an integerl 2, thel-connectivity of a graphG is the minimum number of vertices whose removal fromG produces a disconnected graph with at leastl components or a graph with fewer thanl vertices. A graphG is (n, l)-connected if itsl-connectivity is at leastn. Several sufficient conditions for a graph to be (n, l)-connected are established. IfS is a set ofl( 3) vertices of a graphG, then anS-path ofG is a path between distinct vertices ofS that contains no other vertices ofS. TwoS-paths are said to be internally disjoint if they have no vertices in common, except possibly end-vertices. For a given setS ofl 2 vertices of a graphG, a sufficient condition forG to contain at leastn internally disjointS-paths, each of length at most 2, is established.     

On the Chromatic Number of the Square of the Kneser Graph <Emphasis Type="Italic">K</Emphasis>(2<Emphasis Type="Italic">k</Emphasis>+1, <Emphasis Type="Italic">k</Emphasis>)

The Kneser graph K(n, k) is the graph whose vertices are the k-element subsets of an n-element set, with two vertices adjacent if the sets are disjoint. The chromatic number of the Kneser graph K(n, k) is n–2k+2. Zoltán Füredi raised the question of determining the chromatic number of the square of the Kneser graph, where the square of a graph is the graph obtained by adding edges joining vertices at distance at most 2. We prove that (K²(2k+1, k))4k when k is odd and (K²(2k+1, k))4k+2 when k is even. Also, we use intersecting families of sets to prove lower bounds on (K²(2k+1, k)), and we find the exact maximum size of an intersecting family of 4-sets in a 9-element set such that no two members of the family share three elements.This work was partially supported by NSF grant DMS-0099608Final version received: April 23, 2003     

Clique‐inverse graphs of K3‐free and K4‐free graphs

The clique graph K(G) of a given graph G is the intersection graph of the collection of maximal cliques of G. Given a family ℱ of graphs, the clique‐inverse graphs of ℱ are the graphs whose clique graphs belong to ℱ. In this work, we describe characterizations for clique‐inverse graphs of K₃‐free and K₄‐free graphs. The characterizations are formulated in terms of forbidden induced subgraphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 257–272, 2000     

Equivariant collapses and the homotopy type of iterated clique graphs

The clique graph K(G) of a simple graph G is the intersection graph of its maximal complete subgraphs, and we define iterated clique graphs by K⁰(G)=G, Kⁿ⁺¹(G)=K(Kⁿ(G)). We say that two graphs are homotopy equivalent if their simplicial complexes of complete subgraphs are so. From known results, it can be easily inferred that Kⁿ(G) is homotopy equivalent to G for every n if G belongs to the class of clique-Helly graphs or to the class of dismantlable graphs. However, in both of these cases the collection of iterated clique graphs is finite up to isomorphism. In this paper, we show two infinite classes of clique-divergent graphs that satisfy G?Kⁿ(G) for all n, moreover Kⁿ(G) and G are simple-homotopy equivalent. We provide some results on simple-homotopy type that are of independent interest.     

Iteratedk-line graphs

For integersk≥2, thek-line graph L_k(G) of a graph G is defined as a graph whose vertices correspond to the complete subgraphs onk vertices in G with two distinct vertices adjacent if the corresponding complete subgraphs have 1 common vertices inG. We define iteratedk-line graphs byL _k ⁿ (G) ?L _k (L _k ^n?1 (G), whereL _k ⁰ (G) ?G. In this paper the iterated behavior of thek-line graph operator is investigated. It turns out that the behavior is quite different fork = 2 (the well-known line graph case),k = 3, and k≥4.