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Hadwiger Number and the Cartesian Product of Graphs
 GRAPHS AND COMBINATORICS
, 2008
"... The Hadwiger number η(G) of a graph G is the largest integer n for which the complete graph Kn on n vertices is a minor of G. Hadwiger conjectured that for every graph G, η(G) ≥ χ(G), whereχ(G) is the chromatic number of G. Inthispaper,westudythe Hadwiger number of the Cartesian product G ✷ H of gr ..."
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The Hadwiger number η(G) of a graph G is the largest integer n for which the complete graph Kn on n vertices is a minor of G. Hadwiger conjectured that for every graph G, η(G) ≥ χ(G), whereχ(G) is the chromatic number of G. Inthispaper,westudythe Hadwiger number of the Cartesian product G ✷ H of graphs. As the main result of this paper, we prove that η(G 1 ✷ G 2) ≥ h √ l (1 − o(1)) for any two graphs G 1 and G 2 with η(G 1) = h and η(G 2) = l. We show that the above lower bound is asymptotically best possible when h ≥ l. This asymptotically settles a question of Z. Miller (1978). As consequences of our main result, we show the following: 1. Let G be a connected graph. Let G = G 1 ✷ G 2 ✷... ✷ Gk be the (unique) prime factorization of G.ThenG satisfies Hadwiger’s conjecture if k ≥ 2 log log χ(G) + c ′,wherec ′ is a constant. This improves the 2 log χ(G) + 3 bound in [2]. 2. Let G 1 and G 2 be two graphs such that χ(G 1) ≥ χ(G 2) ≥ c log 1.5 (χ(G 1)),wherec is a constant. Then G 1 ✷ G 2 satisfies Hadwiger’s conjecture. 3. Hadwiger’s conjecture is true for G d (Cartesian product of G taken d times) for every graph G and every d ≥ 2. This settles a question by Chandran and Sivadasan [2]. (They had shown that the Hadiwger’s conjecture is true for G d if d ≥ 3).
Some Results About Minimum Cuts, Treewidth and Hamiltonian Circuits
, 2004
"... Acknowledgements I thank Dr. Ramesh Hariharan for being my advisor and for giving me the freedom to explore whatever I liked. I thank him for the openness of his outlook. I am greatly indebted to Infosys technologies Ltd. for supporting my research via the Infosys Fellowship. I thank Shankar Ram for ..."
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Acknowledgements I thank Dr. Ramesh Hariharan for being my advisor and for giving me the freedom to explore whatever I liked. I thank him for the openness of his outlook. I am greatly indebted to Infosys technologies Ltd. for supporting my research via the Infosys Fellowship. I thank Shankar Ram for being a close friend and companion. He was a great help in surviving the struggle of Ph.D, which involved slowly carving out a new philosophy of existence for oneself from the disheartening mist of disillusionment. I am greatly indebted to my family: My parents and my sisters. If I hadn't taken this path, I could probably have given them much more. But they have never complained... I am thankful to C.R.S and Nari for sharing their ideas with me and for being nice friends and for helping me in many ways.
Triangulations and the Hajós Conjecture
, 2005
"... The Hajós Conjecture was disproved in 1979 by Catlin. Recently, Thomassen showed that there are many ways that Hajós conjecture can go wrong. On the other hand, he observed that locally planar graphs and triangulations of the projective plane and the torus satisfy Hajós Conjecture, and he conjecture ..."
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The Hajós Conjecture was disproved in 1979 by Catlin. Recently, Thomassen showed that there are many ways that Hajós conjecture can go wrong. On the other hand, he observed that locally planar graphs and triangulations of the projective plane and the torus satisfy Hajós Conjecture, and he conjectured that the same holds for arbitrary triangulations of closed surfaces. In this note we disprove the conjecture and show that also in the case of triangulations, there are many reasons why the Hajós Conjecture can fail.