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Treewidth of the q-Kneser graphs

Published: 01 February 2024 Publication History
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  • Abstract

    Let V be an n-dimensional vector space over a finite field F q, where q is a prime power. Define the generalized q -Kneser graph K q ( n, k, t ) to be the graph whose vertices are the k-dimensional subspaces of V and two vertices F 1 and F 2 are adjacent if dim ( F 1 ∩ F 2 ) < t. Then K q ( n, k, 1 ) is the well-known q-Kneser graph. In this paper, we determine the treewidth of K q ( n, k, t ) for n ≥ 2 t ( k − t + 1 ) + k + 1 and t ≥ 1 exactly. Especially, for any possible n, k and q we also determine the treewidth of K q ( n, k, k − 1 ), which is the complement of the Grassmann graph G q ( n, k ).

    References

    [1]
    Arnborg S., Corneil D.G., Proskurowski A., Complexity of finding embeddings in a k-tree, SIAM J. Algebr. Discrete Methods 8 (1987) 277–284.
    [2]
    Blokhuis A., Brouwer A.E., Szőnyi T., On the chromatic number of q-Kneser graphs, Des. Codes Cryptogr. 65 (2012) 187–197.
    [3]
    Bodlaender H.L., A tourist guide through treewidth, Acta Cybernet. 11 (1993) 1–21.
    [4]
    Bodlaender H.L., A linear-time algorithm for finding tree-decompositions of small treewidth, SIAM J. Comput. 25 (1996) 1305–1317.
    [5]
    Brouwer A.E., Cohen A.M., Neumaier A., Distance-Regular Graphs, Springer, Berlin, Heidelberg, New York, 1989.
    [6]
    Erdős P., Ko C., Rado R., Intersection theorems for systems of finite sets, Q. J. Math. Oxf. Ser. 12 (1961) 313–320.
    [7]
    Harvey D.J., Wood D.R., Treewidth of the Kneser graph and the Erdős-Ko-Rado theorem, Electron. J. Combin. 21 (2014) 48–58.
    [8]
    Harvey D.J., Wood D.R., The treewidth of line graphs, J. Combin. Theory Ser. B 132 (2018) 157–179.
    [9]
    Huang L., Lv B., Wang K., Erdős-Ko-Rado theorem, Grassmann graphs and p s-Kneser graphs for vector spaces over a residue class ring, J. Combin. Theory Ser. A 164 (2019) 125–158.
    [10]
    Kloks T., Treewidth of circle graphs, in: Algorithms and Computation (Hong Kong, 1993), in: Lecture Notes in Comput. Sci., vol. 762, Springer, Berlin, 1993, pp. 108–117.
    [11]
    Kneser M., Aufgabe 360, Jahresber. Dtsch. Math.-Ver. 58 (1955) 27.
    [12]
    Li A., Müller T., On the treewidth of random geometric graphs and percolated grids, Adv. Appl. Probab. 49 (2017) 49–60.
    [13]
    Liu K., Cao M., Lu M., Treewidth of the generalized Kneser graphs, Electron. J. Combin. 29 (1) (2022) 19. Paper No. 1.57.
    [14]
    Lovász L., Kneser’s conjecture, chromatic number, and homotopy, J. Combin. Theory Ser. A 25 (1978) 319–324.
    [15]
    Lv B., Wang K., The eigenvalues of q-Kneser graphs, Discrete Math. 312 (2012) 1144–1147.
    [16]
    Lv B., Wang K., The energy of q-Kneser graphs and attenuated q-Kneser graphs, Discrete Appl. Math. 161 (2013) 2079–2083.
    [17]
    Mitsche D., Perarnau G., On treewidth and related parameters of random geometric graphs, SIAM J. Discrete Math. 31 (2017) 1328–1354.
    [18]
    Numata M., A characterization of Grassmann and Johnson graphs, J. Combin. Theory Ser. B 48 (1990) 178–190.
    [19]
    Robertson N., Seymour P., Graph minors. I. Excluding a forest, J. Combin. Theory Ser. B 35 (1983) 39–61.
    [20]
    Robertson N., Seymour P., Graph minors. III. Planar tree-width, J. Combin. Theory Ser. B 36 (1984) 49–64.
    [21]
    Robertson N., Seymour P., Graph minors. II. Algorithmic aspects of tree-width, J. Algorithms 7 (1986) 309–322.
    [22]
    Tanaka H., Classification of subsets with minimal width and dual width in Grassmann, bilinear forms and dual polar graphs, J. Combin. Theory Ser. A 113 (5) (2006) 903–910.
    [23]
    Tanaka H., The Erdős-Ko-Rado theorem for twisted Grassmann graphs, Combinatorica 32 (2012) 735–740.
    [24]
    Wood D.R., Treewidth of Cartesian products of highly connected graphs, J. Graph Theory 73 (2013) 318–321.

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        Published In

        cover image Discrete Applied Mathematics
        Discrete Applied Mathematics  Volume 342, Issue C
        Jan 2024
        398 pages

        Publisher

        Elsevier Science Publishers B. V.

        Netherlands

        Publication History

        Published: 01 February 2024

        Author Tags

        1. Treewidth
        2. Tree decomposition
        3. q-Kneser graph
        4. Grassmann graph

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