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Every graph with chromatic number at least 4 has a subgraph that is a subdivision of K4

A generalization of Dirac's theorem: Subdivisions of wheels

Discrete Mathematics, no. 1 (2005): 202-205

被引用1|浏览10
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摘要

In this paper, we prove that if every vertex of a simple graph has degree at least δ , then it has a subgraph that is isomorphic to a subdivision of a δ -wheel. We then extend a result of Dirac showing that every graph with a chromatic number exceeding n has a subgraph that is a subdivision of the n -wheel.

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简介
  • In 1960, Dirac proved the following theorem [3]. From this theorem, it is easy to derive Corollary 1 which he first published in 1952 [2].
  • Let G be a simple graph with minimum vertex degree .
  • Every graph with chromatic number at least 4 has a subgraph that is a subdivision of K4.
  • If G is a simple graph with 3, G has a subgraph that is a subdivision of W .
重点内容
  • In 1960, Dirac proved the following theorem [3]
  • Let G be a simple graph with minimum vertex degree
  • If 3, G has a subgraph that is a subdivision of K4
  • Every graph with chromatic number at least 4 has a subgraph that is a subdivision of K4
  • If G is a simple graph with 3, G has a subgraph that is a subdivision of W
  • If the chromatic number of G is at least n, G has a subgraph that is a subdivision of Wn−1
结果
  • The authors will use the following technical proposition.
  • Any longest path of a graph contains all of the neighbors of its endvertices.
  • If G is a simple graph with 3 having no W -subdivision, G has a longest path P where the following holds: If x is an endvertex of P and x1, x2, .
  • If z is a vertex on P (x, x2), no neighbor of z lies on P [xl, y].
  • The cycle C formed by the edge zz together with P [z, z ] contains every neighbor of x, except possibly x1.
  • Since P [x1, z] meets C in only one vertex, namely z, it is clear that G has a Wl-subdivision formed by C together with P [x1, z] and the edges incident with x.
  • Let P be the path formed by the edges of P [u, x] together with xx2 and P [x2, y].
  • Since P [u, x] is a path containing u1 but avoiding P [x2, xl], it is clear that G has a Wm-subdivision formed by C together with P [u, x] and the edges incident with u.
  • By Lemma 2, the path P is a longest path with fewer than |E(P [x1, x2])| edges on P [u1, u2]; a contradiction.
  • Let G be a counterexample to the theorem, and let P be a path of G guaranteed by Proposition 1.
  • If y is the other endvertex of P, the path P , formed by the edges of P [y, x2] together with x2x and xx1, is a longest path having x1 as one of its endvertices.
结论
  • Observe that the cycle formed by the edges of P [x2, xl] together with xlx and xx2 contains every neighbor of x1.
  • Theorem 2 guarantees a W -subdivision in a graph with 3.
  • The result, which generalizes Corollary 1, can be derived from Theorem 2.
  • Since x is a vertex of minimum degree, (G) n − 1, and by Theorem 2, it is clear that G has a Wn−1-subdivision.
基金
  • The writing of this article was partially supported by the Louisiana Educational Quality Support Fund under Grant LEQSF(2003-06)-RD-A-19
引用论文
  • J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, North-Holland, New York, 1976.
    Google ScholarFindings
  • G.A. Dirac, The structure of k-chromatic graphs and some remarks on critical graphs, J. Lond. Math. Soc. 27 (1952) 269–271.
    Google ScholarLocate open access versionFindings
  • G.A. Dirac, In abstrakten Graphen vorhandene vollständige 4-Graphen und ihre Unterteilungen, Math. Nachr. 22 (1960) 61–85.
    Google ScholarLocate open access versionFindings
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