Forbidden Structures for Planar Perfect Consecutively Colourable Graphs

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Abstract

A consecutive colouring of a graph is a proper edge colouring with posi- tive integers in which the colours of edges incident with each vertex form an interval of integers. The idea of this colouring was introduced in 1987 by Asratian and Kamalian under the name of interval colouring. Sevast- janov showed that the corresponding decision problem is NP-complete even restricted to the class of bipartite graphs. We focus our attention on the class of consecutively colourable graphs whose all induced subgraphs are consecutively colourable, too. We call elements of this class perfect consecutively colourable to emphasise the conceptual similarity to perfect graphs. Obviously, the class of perfect consecutively colourable graphs is induced hereditary, so it can be characterized by the family of induced forbidden graphs. In this work we give a necessary and sufficient conditions that must be satisfied by the generalized Sevastjanov rosette to be an induced forbid- den graph for the class of perfect consecutively colourable graphs. Along the way, we show the exact values of the deficiency of all generalized Sevastjanov rosettes, which improves the earlier known estimating result. It should be mentioned that the deficiency of a graph measures its closeness to the class of consecutively colourable graphs. We motivate the investigation of graphs considered here by showing their connection to the class of planar perfect consecutively colourable graphs.

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  • [1] A.S. Asratian and C.J. Casselgren On interval edge colorings of (αß)-biregular bipartite graphs Discrete Math. 307 (2006) 1951-1956. doi:

    • Crossref
    • Export Citation
  • [2] A.S. Asratian and R.R. Kamalian Interval colorings of the edges of multigraph Appl. Math. 5 (1987) 25-34 in Russian.

  • [3] A.S. Asratian and R.R. Kamalian Investigation on interval edge-colorings of graphs J. Combin. Theory Ser. B 62 (1994) 34-43. doi:

    • Crossref
    • Export Citation
  • [4] M.A. Axenovich On interval colorings of planar graphs Congr. Numer. 159 (2002) 77-94.

  • [5] M. Borowiecka-Olszewska E. Drgas-Burchardt and M. Ha luszczak On the structure and deficiency of k-trees with bounded degree Discrete Appl. Math. 201 (2016) 24-37. doi:

    • Crossref
    • Export Citation
  • [6] M. Borowiecka-Olszewska and E. Drgas-Burchardt The deficiency of all generalized Hertz graphs and minimal consecutively non-colourable graphs in this class Discrete Math. 339 (2016) 1892-1908. doi:

    • Crossref
    • Export Citation
  • [7] R. Diestel Graph Theory 2nd ed. (Graduate Texts in Mathematics 173 Springer-Verlag New York 2000).

  • [8] K. Giaro The complexity of consecutive _-coloring of bipartite graphs: 4 is easy 5 is hard Ars Combin. 47 (1997) 287-300.

  • [9] K. Giaro and M. Kubale Consecutive edge-colorings of complete and incomplete Cartesian products of graphs Congr. Numer. 128 (1997) 143-149.

  • [10] K. Giaro and M. Kubale Compact scheduling of zero-one time operations in multi-stage systems Discrete Appl. Math. 145 (2004) 95-103. doi:

    • Crossref
    • Export Citation
  • [11] K. Giaro M. Kubale and M. Ma lafiejski On the deficiency of bipartite graphs Discrete Appl. Math. 94 (1999) 193-203. doi:

    • Crossref
    • Export Citation
  • [12] D.L. Greenwell R.L. Hemminger and J. Klerlein Forbidden subgraphs in: Proc. Of the 4th S-E Conf. Combinatorics Graph Theory and Computing (Utilitas Math. Winnipeg Man. 1973) 389-394.

  • [13] D. Hanson and C.O.M. Loten A lower bound for interval colouring of bi-regular bipartite graphs Bull. Inst. Comb. Appl. 18 (1996) 69-74.

  • [14] D. Hanson C.O.M. Loten and B. Toft On interval colorings of bi-regular bipartite graphs Ars Combin. 50 (1998) 23-32.

  • [15] R.R. Kamalian Interval Coloring of Complete Bipartite Graphs and Trees (in Russian) (Preprint of Comp. Cen. of Acad. Sci. of Armenian SSR Erevan 1989).

  • [16] M. Kubale Graph Colorings (American Mathematical Society Providence Rhode Island 2004). doi:

    • Crossref
    • Export Citation
  • [17] P.A. Petrosyan Interval edge-colorings of complete graphs and n-dimensional cubes Discrete Math. 310 (2010) 1580-1587. doi:

    • Crossref
    • Export Citation
  • [18] P.A. Petrosyan Interval edge colorings of some products of graphs Discuss. Math. Graph Theory 31 (2011) 357-373. doi:

    • Crossref
    • Export Citation
  • [19] P.A. Petrosyan H.H. Khachatrian and H.G. Tananyan Interval edge-colorings of Cartesian products of graphs I Discuss. Math. Graph Theory 33 (2013) 613-632. doi:

    • Crossref
    • Export Citation
  • [20] P.A. Petrosyan and H.H. Khachatrian Interval non-edge-colorable bipartite graphs and multigraphs J. Graph Theory 76 (2014) 200-216. doi:

    • Crossref
    • Export Citation
  • [21] A.V. Pyatkin Interval coloring of (3 4)-biregular bipartite graphs having large cubic subgraphs J. Graph Theory 47 (2004) 122-128. doi:

    • Crossref
    • Export Citation
  • [22] S.V. Sevastjanov On interval colorability of bipartite graphs Met. Diskret Analiz. 50 (1990) 61-72 in Russian.

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Target audience:

researchers in the fields of: colourings, partitions (general colourings), hereditary properties, independence and dominating structures (sets, paths, cycles, etc.), cycles, local properties, products of graphs

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