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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

EI

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.

• In 1960, Dirac proved the following theorem . From this theorem, it is easy to derive Corollary 1 which he first published in 1952 .
• 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 .

• In 1960, Dirac proved the following theorem 
• 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

• 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.

• 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.

• The writing of this article was partially supported by the Louisiana Educational Quality Support Fund under Grant LEQSF(2003-06)-RD-A-19 0 