Maximal independent sets in caterpillar graphs

A caterpillar graph is a tree in which the removal of all pendant vertices results in a chordless path. In this work, we determine the number of maximal independent sets (mis) in caterpillar graphs. For a general graph, this problem is #P—complete. We provide a polynomial time algorithm to generate the whole family of mis in a caterpillar graph. We also characterize the independent graph (intersection graph of mis) and the clique graph (intersection graph of cliques) of complete caterpillar graphs.     

Separator theorems and Turán-type results for planar intersection graphs

We establish several geometric extensions of the Lipton-Tarjan separator theorem for planar graphs. For instance, we show that any collection C of Jordan curves in the plane with a total of m crossings has a partition into three parts C=S∪C₁∪C₂ such that , , and no element of C₁ has a point in common with any element of C₂. These results are used to obtain various properties of intersection patterns of geometric objects in the plane. In particular, we prove that if a graph G can be obtained as the intersection graph of n convex sets in the plane and it contains no complete bipartite graph Kt,t as a subgraph, then the number of edges of G cannot exceed ctn, for a suitable constant ct.     

The maximum number of maximal independent sets in unicyclic connected graphs

In this paper, we determine the maximum number of maximal independent sets in a unicyclic connected graph. We also find a class of graphs achieving this maximum value.     

Connected graphs without long paths

We determine the maximum number of edges in a connected graph with n vertices if it contains no path with k+1 vertices. We also determine the extremal graphs.     

New families of graphs without short cycles and large size

We denote by ex(n;{C₃,C₄,…,C_s}) or f_s(n) the maximum number of edges in a graph of order n and girth at least s+1. First we give a method to transform an n-vertex graph of girth g into a graph of girth at least g−1 on fewer vertices. For an infinite sequence of values of n and s∈{4,6,10} the obtained graphs are denser than the known constructions of graphs of the same girth s+1. We also give another different construction of dense graphs for an infinite sequence of values of n and s∈{7,11}. These two methods improve the known lower bounds on f_s(n) for s∈{4,6,7,10,11} which were obtained using different algorithms. Finally, to know how good are our results, we have proved that for s∈{5,7,11}, and for s∈{6,10}.     

Regular bipartite graphs are antimagic

A labeling of a graph G is a bijection from E(G) to the set {1, 2,… |E(G)|}. A labeling is antimagic if for any distinct vertices u and v, the sum of the labels on edges incident to u is different from the sum of the labels on edges incident to v. We say a graph is antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that every connected graph other than K₂ is antimagic. In this article, we show that every regular bipartite graph (with degree at least 2) is antimagic. Our technique relies heavily on the Marriage Theorem. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 173–182, 2009     

Independent sets in regular graphs and sum-free subsets of finite groups

It is shown that there exists a function∈(k) which tends to 0 ask tends to infinity, such that anyk-regular graph onn vertices contains at most 2^{(1/2+∈(k))n} independent sets. This settles a conjecture of A. Granville and has several applications in Combinatorial Group Theory. Research supported in part by the United States-Israel Binational Science Foundation and by a Bergmann Memorial Grant.     

Fractional clique decompositions of dense graphs

For each , we show that any graph G with minimum degree at least has a fractional K_r‐decomposition. This improves the best previous bounds on the minimum degree required to guarantee a fractional K_r‐decomposition given by Dukes (for small r) and Barber, Kühn, Lo, Montgomery, and Osthus (for large r), giving the first bound that is tight up to the constant multiple of r (seen, for example, by considering Turán graphs). In combination with work by Glock, Kühn, Lo, Montgomery, and Osthus, this shows that, for any graph F with chromatic number , and any , any sufficiently large graph G with minimum degree at least has, subject to some further simple necessary divisibility conditions, an (exact) F‐decomposition.     

On sets of integers whose shifted products are powers

Let N be a positive integer and let A be a subset of {1,…,N} with the property that aa^′+1 is a pure power whenever a and a^′ are distinct elements of A. We prove that |A|, the cardinality of A, is not large. In particular, we show that |A|?(logN)^2/3(loglogN)^1/3.     

Spanning 3-colourable subgraphs of small bandwidth in dense graphs

A conjecture by Bollobás and Komlós states the following: For every γ>0 and integers r2 and Δ, there exists β>0 with the following property. If G is a sufficiently large graph with n vertices and minimum degree at least and H is an r-chromatic graph with n vertices, bandwidth at most βn and maximum degree at most Δ, then G contains a copy of H.This conjecture generalises several results concerning sufficient degree conditions for the containment of spanning subgraphs. We prove the conjecture for the case r=3.     

Maximize the Q-index of graphs with fixed order and size

In this paper, we consider two kinds of spectral extremal questions. The first asks which graph attains the maximum Q-index over all graphs of order n and size $m=n+k$? The second asks which graph attains the maximum Q-index over all $(p,q)$-bipartite graphs with $m=p+q+k$ edges? We solve the first question for $4\le k\le n?3$, and the second question for $?p?q\le k\le p?q$. The maximum Q-index on connected $(p,q)$-bipartite graphs is also determined for $?1\le k\le p?2$.     

-graphs: An efficient tool for constructing symmetric and semisymmetric graphs

Symmetric and semisymmetric graphs are used in many scientific domains, especially parallel computation and interconnection networks. The industry and the research world make a huge usage of such graphs. Constructing symmetric and semisymmetric graphs is a large and hard problem. In this paper a tool called G-graphs and based on group theory is used. We show the efficiency of this tool for constructing symmetric and semisymmetric graphs and we exhibit experimental results.     

Integral circulant graphs

In this note we characterize integral graphs among circulant graphs. It is conjectured that there are exactly 2^τ(n)-1 non-isomorphic integral circulant graphs on n vertices, where τ(n) is the number of divisors of n.     

关于带宽极值问题的两个结果

本文研究的问题是确定ｅ＊（ｐ,Ｂ）的值,也就是确定顶点数为ｐ、带宽为Ｂ的连通图Ｇ的最小边数,本文给出当Ｂ＝ｐ＋３／２和Ｂ＝ｐ／２＋２时的精确结果。