Results 1  10
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32
The Colin de Verdière graph parameter
, 1997
"... In 1990, Y. Colin de Verdière introduced a new graph parameter (G), based on spectral properties of matrices associated with G. He showed that (G) is monotone under taking minors and that planarity of G is characterized by the inequality (G) 3. Recently Lovasz and Schrijver showed that linkless emb ..."
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Cited by 16 (1 self)
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In 1990, Y. Colin de Verdière introduced a new graph parameter (G), based on spectral properties of matrices associated with G. He showed that (G) is monotone under taking minors and that planarity of G is characterized by the inequality (G) 3. Recently Lovasz and Schrijver showed that linkless embeddability of G is characterized by the inequality (G) 4. In this paper we give an overview of results on (G) and of techniques to handle it.
Relaxing planarity for topological graphs
 Discrete and Computational Geometry, Lecture Notes in Comput. Sci., 2866
, 2003
"... Abstract. According to Euler’s formula, every planar graph with n vertices has at most O(n) edges. How much can we relax the condition of planarity without violating the conclusion? After surveying some classical and recent results of this kind, we prove that every graph of n vertices, which can be ..."
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Cited by 6 (3 self)
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Abstract. According to Euler’s formula, every planar graph with n vertices has at most O(n) edges. How much can we relax the condition of planarity without violating the conclusion? After surveying some classical and recent results of this kind, we prove that every graph of n vertices, which can be drawn in the plane without three pairwise crossing edges, has at most O(n) edges. For straightline drawings, this statement has been established by Agarwal et al., using a more complicated argument, but for the general case previously no bound better than O(n 3/2) was known. 1
Graph Color Extensions: When Hadwiger's Conjecture and Embeddings Help
 Electronic J. Comb
, 2002
"... Suppose G is rcolorable and P V (G) is such that the components of G[P ] are far apart. We show that any (r + s)coloring of G[P ] in which each component is scolored extends to an (r + s)coloring of G. If G does not contract to K 5 or is planar and s 2, then any (r + s 1)coloring of P in w ..."
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Cited by 5 (1 self)
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Suppose G is rcolorable and P V (G) is such that the components of G[P ] are far apart. We show that any (r + s)coloring of G[P ] in which each component is scolored extends to an (r + s)coloring of G. If G does not contract to K 5 or is planar and s 2, then any (r + s 1)coloring of P in which each component is scolored extends to an (r + s 1)coloring of G. This result uses the Four Color Theorem and its equivalence to Hadwiger's Conjecture for k = 5. For s = 2 this provides an armative answer to a question of Thomassen. Similar results hold for coloring arbitrary graphs embedded in both orientable and nonorientable surfaces.
How the proof of the strong perfect graph conjecture was found
, 2006
"... In 1961, Claude Berge proposed the “strong perfect graph conjecture”, probably the most beautiful open question in graph theory. It was answered just before his death in 2002. This is an overview of the solution, together with an account of some of the ideas that eventually brought us to the answer. ..."
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Cited by 3 (0 self)
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In 1961, Claude Berge proposed the “strong perfect graph conjecture”, probably the most beautiful open question in graph theory. It was answered just before his death in 2002. This is an overview of the solution, together with an account of some of the ideas that eventually brought us to the answer.
Complete Minors, Independent Sets, and Chordal Graphs
"... Abstract. The Hadwiger number h(G) of a graph G is the maximum size of a complete minor of G. Hadwiger’s Conjecture states that h(G) ≥ χ(G). Since χ(G)α(G) ≥ V (G), Hadwiger’s Conjecture implies that α(G)h(G) ≥ V (G). We show that (2α(G) − ⌈log τ (τα(G)/2)⌉)h(G) ≥ V (G)  where τ ≈ 6.83. Fo ..."
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Cited by 3 (1 self)
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Abstract. The Hadwiger number h(G) of a graph G is the maximum size of a complete minor of G. Hadwiger’s Conjecture states that h(G) ≥ χ(G). Since χ(G)α(G) ≥ V (G), Hadwiger’s Conjecture implies that α(G)h(G) ≥ V (G). We show that (2α(G) − ⌈log τ (τα(G)/2)⌉)h(G) ≥ V (G)  where τ ≈ 6.83. For graphs with α(G) ≥ 14, this improves on a recent result of Kawarabayashi and Song who showed (2α(G) − 2)h(G) ≥ V (G)  when α(G) ≥ 3.
Some Remarks on the Odd Hadwiger's Conjecture
, 2005
"... We say that H has an odd complete minor of order at least l if there are l vertex disjoint trees in H such that every two of them are joined by an edge, and in addition, all the vertices of trees are twocolored in such a way that the edges within the trees are bichromatic, but the edges between tre ..."
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Cited by 3 (3 self)
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We say that H has an odd complete minor of order at least l if there are l vertex disjoint trees in H such that every two of them are joined by an edge, and in addition, all the vertices of trees are twocolored in such a way that the edges within the trees are bichromatic, but the edges between trees are monochromatic. Gerards and Seymour conjectured that if a graph has no odd complete minor of order l, then it is (l − 1)colorable. This is substantially stronger than the wellknown conjecture of Hadwiger. Recently, Geelen et al. proved that there exists a constant c such that any graph with no odd Kkminor is ck √ log kcolorable. However, it is not known if there exists an absolute constant c such that any graph with no odd Kkminor is ckcolorable.
Reducing Haj'os ' coloring conjecture to 4connected graphs
, 2004
"... Abstract Haj'os conjectured that, for any positive integer k, every graph containing no Kk+1subdivision is kcolorable. This is true when k < = 3, and false when k> = 6.Haj'os ' conjecture remains open for k = 4, 5. In this paper, we show that any possiblecounterexample to this conjecture for k = 4 ..."
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Cited by 2 (2 self)
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Abstract Haj'os conjectured that, for any positive integer k, every graph containing no Kk+1subdivision is kcolorable. This is true when k < = 3, and false when k> = 6.Haj'os ' conjecture remains open for k = 4, 5. In this paper, we show that any possiblecounterexample to this conjecture for k = 4 with minimum number of vertices mustbe 4connected. This is a step in an attempt to reduce Haj'os ' conjecture for k = 4to the conjecture of Seymour that any 5connected nonplanar graph contains a K5subdivision.