On joins of a clique and a co-clique as star complements in regular graphs

Journal of Algebraic Combinatorics - In this paper we consider r-regular graphs G that admit the vertex set partition such that one of the induced subgraphs is the join of an s-vertex clique and a...     

Co-cliques and star complements in extremal strongly regular graphs

Suppose that the positive integer μ is the eigenvalue of largest multiplicity in an extremal strongly regular graph G. By interlacing, the independence number of G is at most 4μ² + 4μ − 2. Star complements are used to show that if this bound is attained then either (a) μ = 1 and G is the Schläfli graph or (b) μ = 2 and G is the McLaughlin graph.     

Maximum matchings in regular graphs

It was conjectured by Mkrtchyan, Petrosyan and Vardanyan that every graph $G$ with $\Delta \left(G\right)?\delta \left(G\right)\le 1$ has a maximum matching $M$ such that any two $M$-unsaturated vertices do not share a neighbor. The results obtained in Mkrtchyan et al. (2010), Petrosyan (2014) and Picouleau (2010) leave the conjecture unknown only for $k$-regular graphs with $4\le k\le 6$. All counterexamples for $k$-regular graphs $(k\ge 7)$ given in Petrosyan (2014) have multiple edges. In this paper, we confirm the conjecture for all $k$-regular simple graphs and also $k$-regular multigraphs with $k\le 4$.     

Random perfect matchings in regular graphs

We prove that in all regular robust expanders $G$, every edge is asymptotically equally likely contained in a uniformly chosen perfect matching $M$. We also show that given any fixed matching or spanning regular graph $N$ in $G$, the random variable $|M\cap E(N)|$ is approximately Poisson distributed. This in particular confirms a conjecture and a question due to Spiro and Surya, and complements results due to Kahn and Kim who proved that in a regular graph every vertex is asymptotically equally likely contained in a uniformly chosen matching. Our proofs rely on the switching method and the fact that simple random walks mix rapidly in robust expanders.     

Perfect matchings in regular bipartite graphs

P. Hall, [2], gave necessary and sufficient conditions for a bipartite graph to have a perfect matching. Koning, [3], proved that such a graph can be decomposed intok edge-disjoint perfect matchings if and only if it isk-regular. It immediately follows that in ak-regular bipartite graphG, the deletion of any setS of at mostk – 1 edges leaves intact one of those perfect matchings. However, it is not known what happens if we delete more thank – 1 edges. In this paper we give sufficient conditions so that by deleting a setS ofk + r edgesr 0, stillG – S has a perfect matching. Furthermore we prove that our result, in some sense, is best possible.     

Maximum induced matchings in graphs

We provide a formula for the number of edges of a maximum induced matching in a graph. As applications, we give some structural properties of (k + 1)K₂-free graphs, construct all 2K₂-free graphs, and count the number of labeled 2K₂-free connected bipartite graphs.     

Improved induced matchings in sparse graphs

An induced matching in a graph G=(V,E) is a matching M such that (V,M) is an induced subgraph of G. Clearly, among two vertices with the same neighbourhood (called twins) at most one is matched in any induced matching, and if one of them is matched then there is another matching of the same size that matches the other vertex. Motivated by this, Kanj et al. [10] studied induced matchings in twinless graphs. They showed that any twinless planar graph contains an induced matching of size at least and that there are twinless planar graphs that do not contain an induced matching of size greater than . We improve both these bounds to , which is tight up to an additive constant. This implies that the problem of deciding whether a planar graph has an induced matching of size k has a kernel of size at most 28k. We also show for the first time that this problem is fixed parameter tractable for graphs of bounded arboricity.Kanj et al. also presented an algorithm which decides in -time whether an n-vertex planar graph contains an induced matching of size k. Our results improve the time complexity analysis of their algorithm. However, we also show a more efficient -time algorithm. Its main ingredient is a new, O^∗(4^l)-time algorithm for finding a maximum induced matching in a graph of branch width at most l.     

The rainbow number of matchings in regular bipartite graphs

Given a graph G and a subgraph H of G, let rb(G,H) be the minimum number r for which any edge-coloring of G with r colors has a rainbow subgraph H. The number rb(G,H) is called the rainbow number of H with respect to G. Denote as mK₂ a matching of size m and as B_n,k the set of all the k-regular bipartite graphs with bipartition (X,Y) such that X=Y=n and k≤n. Let k,m,n be given positive integers, where k≥3, m≥2 and n>3(m−1). We show that for every GB_n,k, rb(G,mK₂)=k(m−2)+2. We also determine the rainbow numbers of matchings in paths and cycles.     

On disjoint matchings in cubic graphs

For $i=2,3$ and a cubic graph $G$ let ${\nu }_i\left(G\right)$ denote the maximum number of edges that can be covered by $i$ matchings. We show that ${\nu }_2\left(G\right)\ge \frac{4}{5}\left|V\left(G\right)\right|$ and ${\nu }_3\left(G\right)\ge \frac{7}{6}\left|V\left(G\right)\right|$. Moreover, it turns out that ${\nu }_2\left(G\right)\le \frac{\left|V\left(G\right)\right|+2{\nu }_3\left(G\right)}{4}$.     

On induced matchings

Let q^*(G) denote the minimum integer t for which E(G) can be partitioned into t induced matchings of G. Faudree et al. conjectured that q^*(G)d², if G is a bipartite graph and d is the maximum degree of G. In this note, we give an affirmative answer for d=3, the first nontrivial case of this conjecture.     

Star complements in regular graphs: Old and new results

We survey results concerning star complements in finite regular graphs, and note the connection with designs and strongly regular graphs in certain cases. We include improved proofs along with new results on stars and windmills as star complements.     

On cuts and matchings in planar graphs

We study the max cut problem in graphs not contractible toK ₅, and optimum perfect matchings in planar graphs. We prove that both problems can be formulated as polynomial size linear programs.Supported by the joint project Combinatorial Optimization of the Natural Sciences and Engineering Research Council of Canada and the German Research Association (Deutsche Forschungsgemeinschaft, SFB 303).