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WANG Xiao. The chromatic number for fork-free graphs[J]. Journal of East China Normal University (Natural Sciences), 2016, (1): 102-106. doi: 10.3969/j.issn.1000-5641.2016.01.0051
Citation: WANG Xiao. The chromatic number for fork-free graphs[J]. Journal of East China Normal University (Natural Sciences), 2016, (1): 102-106. doi: 10.3969/j.issn.1000-5641.2016.01.0051

The chromatic number for fork-free graphs

doi: 10.3969/j.issn.1000-5641.2016.01.0051
  • Received Date: 2014-10-29
  • Publish Date: 2016-01-25
  • Randerath once conjectured that every triangle-free and fork-free graph is 3-colourable. By a lemma, the conjecture for C_4-free graphs was proved.Moreover, the result that every triangle-free, C_4-free and C_{2,2,1,n}-free graph is (n+2)-colourable was proved as well, where C_{2,2,1,n} is the long handled fork with order (n+6) obtained from E-graph and P_n by joining the center vertex of E and one endvertex of P_n.
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    [2]GY\'{A]RF\'{A]S A. Problems from the world surrounding perfect graphs [J]. Zastow Mat, 1987, 19: 413-441.
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    [4]GY\'{A]RF\'{A]S A, SZEMER\'{E]DI E and Tuza. Induced subtrees ingraphs of large chromatic number [J]. Discrete Math, 1980, 30(3):235-244.
    [5] DUAN F and WU B Y. On chromatic number of graphs without certain induced subgraph [J]. Ars combinatoria, 2011, 101: 33-34.
    [6] DUAN F and ZHANG W J. On chromatic number of $\{2K_1+K_2, C_4\]$-free graphs [J]. Journal of East China Normal University: Natural Science, 2014(1): 9-12.
    [7] RANDERATH B, SCHIERMEYER I.  A note on Brooks' theorem for triangle-free graphs [J]. Australas J Comb, 2002, 26: 3-9.
    [8] RANDERATH B, SCHIERMEYER I. Vertex coloring and forbidden subgraphs-a survey [J]. Graphs Combin, 2004, 20(1): 1-40.
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    [10] FAN G, XU B, YE T, et al. Forbidden subgraphs and 3-colorings [J]. Siam J Disc Math, 2014, 28: 1226-1256.
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