Strengthening the Lovász bound for graph coloring

The Lovász -number is a way to approximate the independence number of a graph, but also its chromatic number. We express the Lovász bound as the continuous relaxation of a discrete Lovász -number which we derive from Karger et al.s formulation, and which is equal to the chromatic number. We also give another relaxation à la Schrijver-McEliece, which is better than the Lovász -number.     

On the Lovász theta function and some variants

The Lovász theta function of a graph is a well-known upper bound on the stability number. It can be computed efficiently by solving a semidefinite program (SDP). Actually, one can solve either of two SDPs, one due to Lovász and the other to Grötschel et al. The former SDP is often thought to be preferable computationally, since it has fewer variables and constraints. We derive some new results on these two equivalent SDPs. The surprising result is that, if we weaken the SDPs by aggregating constraints, or strengthen them by adding cutting planes, the equivalence breaks down. In particular, the Grötschel et al. scheme typically yields a stronger bound than the Lovász one.     

Combinatorial Groupoids,Cubical Complexes,and the Lovász Conjecture

This paper lays the foundation for a theory of combinatorial groupoids that allows us to use concepts like “holonomy”, “parallel transport”, “bundles”, “combinatorial curvature”, etc. in the context of simplicial (polyhedral) complexes, posets, graphs, polytopes and other combinatorial objects. We introduce a new, holonomy-type invariant for cubical complexes, leading to a combinatorial “Theorema Egregium” for cubical complexes that are non-embeddable into cubical lattices. Parallel transport of Hom-complexes and maps is used as a tool to extend Babson–Kozlov–Lovász graph coloring results to more general statements about nondegenerate maps (colorings) of simplicial complexes and graphs. The author was supported by grants 144014 and 144026 of the Serbian Ministry of Science and Technology.     

Lovász–Schrijver SDP-operator,near-perfect graphs and near-bipartite graphs

We study the Lovász–Schrijver lift-and-project operator (\({{\mathrm{\text {LS}}}}_+\)) based on the cone of symmetric, positive semidefinite matrices, applied to the fractional stable set polytope of graphs. The problem of obtaining a combinatorial characterization of graphs for which the \({{\mathrm{\text {LS}}}}_+\)-operator generates the stable set polytope in one step has been open since 1990. We call these graphs \({{{\mathrm{\text {LS}}}}}_+\)-perfect. In the current contribution, we pursue a full combinatorial characterization of \({{{\mathrm{\text {LS}}}}}_+\)-perfect graphs and make progress towards such a characterization by establishing a new, close relationship among \({{{\mathrm{\text {LS}}}}}_+\)-perfect graphs, near-bipartite graphs and a newly introduced concept of full-support-perfect graphs.     

A remark on the conjecture of Erdös, Faber and Lovász

The celebrated Erd?s, Faber and Lovász Conjecture may be stated as follows: Any linear hypergraph on ν points has chromatic index at most ν. We show that the conjecture is equivalent to the following assumption: For any graph , where ν(G) denotes the linear intersection number and χ(G) denotes the chromatic number of G. As we will see for any graph G = (V, E), where denotes the complement of G. Hence, at least G or fulfills the conjecture.      

On Lovász’ lattice reduction and the nearest lattice point problem

Answering a question of Vera Sós, we show how Lovász’ lattice reduction can be used to find a point of a given lattice, nearest within a factor ofc ^d (c = const.) to a given point in R ^d . We prove that each of two straightforward fast heuristic procedures achieves this goal when applied to a lattice given by a Lovász-reduced basis. The verification of one of them requires proving a geometric feature of Lovász-reduced bases: ac ₁ ^d lower bound on the angle between any member of the basis and the hyperplane generated by the other members, wherec ₁ = √2/3. As an application, we obtain a solution to the nonhomogeneous simultaneous diophantine approximation problem, optimal within a factor ofC ^d . In another application, we improve the Grötschel-Lovász-Schrijver version of H. W. Lenstra’s integer linear programming algorithm. The algorithms, when applied to rational input vectors, run in polynomial time.     

Spectral Characterizations of the Lovász Number and the Delsarte Number of a Graph

This paper gives spectral characterizations of two closely related graph functions: the Lovász number and a generalization ¹ of Delsarte's linear programming bound. There are many known characterizations of the Lovász number , and each one corresponds to a similar characterization of ¹ obtained by extremizing over a larger or smaller class of objects.The spectral characterizations of and ¹ given here involve the largest eigenvalue of a type of weighted Laplacian that Fan Chung introduced.     

Approximating the independence number via theϑ-function

We describe an approximation algorithm for the independence number of a graph. If a graph onn vertices has an independence numbern/k + m for some fixed integerk 3 and somem > 0, the algorithm finds, in random polynomial time, an independent set of size , improving the best known previous algorithm of Boppana and Halldorsson that finds an independent set of size (m ^1/(k–1)) in such a graph. The algorithm is based on semi-definite programming, some properties of the Lovász-function of a graph and the recent algorithm of Karger, Motwani and Sudan for approximating the chromatic number of a graph. If the-function of ann vertex graph is at leastMn ^1–2/k for some absolute constantM, we describe another, related, efficient algorithm that finds an independent set of sizek. Several examples show the limitations of the approach and the analysis together with some related arguments supply new results on the problem of estimating the largest possible ratio between the-function and the independence number of a graph onn vertices. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Research supported in part by a USA—Israel BSF grant and by the Fund for Basic Research administered by the Israel Academy of Sciences and by the Minkowski Minerva Center for Geometry at Tel Aviv University.This work was partly done while the author was at XEROX PARC and partly at DIMACS.     

Chromatic Gallai identities operating on Lovász number

If $G$ is a triangle-free graph, then two Gallai identities can be written as $\alpha (G)+\overline{\chi }(L(G))=|V(G)|=\alpha (L(G))+\overline{\chi }(G)$ , where $\alpha $ and $\overline{\chi }$ denote the stability number and the clique-partition number, and $L(G)$ is the line graph of  $G$ . We show that, surprisingly, both equalities can be preserved for any graph $G$ by deleting the edges of the line graph corresponding to simplicial pairs of adjacent arcs, according to any acyclic orientation of  $G$ . As a consequence, one obtains an operator $\Phi $ which associates to any graph parameter $\beta $ such that $\alpha (G) \le \beta (G) \le \overline{\chi }(G)$ for all graph $G$ , a graph parameter $\Phi _\beta $ such that $\alpha (G) \le \Phi _\beta (G) \le \overline{\chi }(G)$ for all graph $G$ . We prove that $\vartheta (G) \le \Phi _\vartheta (G)$ and that $\Phi _{\overline{\chi }_f}(G)\le \overline{\chi }_f(G)$ for all graph  $G$ , where $\vartheta $ is Lovász theta function and $\overline{\chi }_f$ is the fractional clique-partition number. Moreover, $\overline{\chi }_f(G) \le \Phi _\vartheta (G)$ for triangle-free $G$ . Comparing to the previous strengthenings $\Psi _\vartheta $ and $\vartheta ^{+ \triangle }$ of $\vartheta $ , numerical experiments show that $\Phi _\vartheta $ is a significant better lower bound for $\overline{\chi }$ than $\vartheta $ .     

关于Lovász不等式的几类超图

本文证明了下面的定理:若超图H=(X;E_1,E_2,…,E_m)中存在浓度为k长为m的圈,则有     

Edge coloring of hypergraphs and a conjecture of ErdÖs,Faber, Lovász

Call a bypergraphsimple if for any pairu, v of distinct vertices, there is at most one edge incident to bothu andv, and there are no edges incident to exactly one vertex. A conjecture of Erds, Faber and Lovász is equivalent to the statement that the edges of any simple hypergraph onn vertices can be colored with at mostn colors. We present a simple proof that the edges of a simple hypergraph onn vertices can be colored with at most [1.5n-2 colors].This research was partially supported by N.S.F. grant No. MCS-8311422.     

关于L.Lovász定理的简证

In 1983 J. A. Bondy mentioned a result of L.Lovsz. He said that it seems"magic". Now we give a simple proof of this result by using the basic probability theory only, and we may see why this result holds clearly. Let G be a graph with a specified vertex r.A path P=v_1v_2…v_k is rooted at r if v_1=r.For 1≤j≤k, the number of edges of G which are incident with v_j but with no v_i, i相似文献