New bounds for the max-k-cut and chromatic number of a graph

Abstract

We consider several semidefinite programming relaxations for the max-k-cut problem, with increasing complexity. The optimal solution of the weakest presented semidefinite programming relaxation has a closed form expression that includes the largest Laplacian eigenvalue of the graph under consideration. This is the first known eigenvalue bound for the max-k-cut when k > 2 that is applicable to any graph. This bound is exploited to derive a new eigenvalue bound on the chromatic number of a graph. For regular graphs, the new bound on the chromatic number is the same as the well-known Hoffman bound; however, the two bounds are incomparable in general.We prove that the eigenvalue bound for the max-k-cut is tight for several classes of graphs. We investigate the presented bounds for specific classes of graphs, such as walk-regular graphs, strongly regular graphs, and graphs from the Hamming association scheme.

The smallest eigenvalues of Hamming graphs, Johnson graphs and other distance-regular graphs with classical parameters

“…Cases with three eigenvalues (the connected graphs among these are strongly regular-we give the standard parameters (v, k, λ, µ)): d q j comment 4 2 2 2 copies of 4K 2 5 2 2 2 copies of the Clebsch graph 5 2 4 2 copies of the complement of the Clebsch graph 7 2 4 2 copies of V O + (6, 2) 4 3 2 (81, 24,9,6) More generally, if we take the Hamming scheme H(d, q) with q = 4, and call two distinct vertices adjacent if their distance is even, we obtain a strongly regular graph (as was observed in [19, Case III]), namely the graph V O ± (2d, 2), where the sign is (−1) d . Indeed, the weight of a quaternary digit is given by the (elliptic) binary quadratic form x 2 1 + x 1 x 2 + x 2 2 .…”

“…These graphs provide examples where the performance ratio of the Goemans-Williamson algorithm is tight [1]. The smallest eigenvalues are also used for determining the max-k-cut [6] and the chromatic number of the graphs in the Hamming scheme [6].…”

Projection results for the k-partition problem

“…By definition, any valid inequality for P xy (K n , k) that involves only y variables is valid also for P y (K n , k). This includes not only the inequalities (14), (15), but also inequalities (8), (10) and (11). Still more inequalities for P y (K n , k) are presented in [8,11].…”

“…Finally, we mention two recent papers. Van Dam & Sotirov [10] examine an SDP relaxation whose solution has a closed form (involving Eigenvalues). De Sousa et al [34] show empirically that odd wheel and odd bicycle wheel inequalities can be useful for strengthening SDP relaxations.…”

Computational study of valid inequalities for the maximum k-cut problem

“…Using a similar approach as [32], the authors in [34] propose several new bounds for max-k-cut from the maximum eigenvalue of the Laplacian matrix of G. Moreover, [34] shows that certain perturbations in the diagonal of the Laplacian matrix can lead to stronger bounds. In [26] the author proposes a more robust bound than the one proposed in [34] by using the smallest eigenvalue of the adjacency matrix of G.…”