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Minors in Graphs of Large Girth
 J. Combin. Theory B
, 1988
"... We show that for every odd integer g 5 there exists a constant c such that every graph of minimum degree r and girth at least g contains a minor of minimum degree at least cr . This is best possible up to the value of the constant c for g = 5; 7 and 11. More generally, a wellknown conjecture ..."
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We show that for every odd integer g 5 there exists a constant c such that every graph of minimum degree r and girth at least g contains a minor of minimum degree at least cr . This is best possible up to the value of the constant c for g = 5; 7 and 11. More generally, a wellknown conjecture about the minimal order of graphs of given minimum degree and large girth would imply that our result gives the correct order of magnitude for all odd values of g. The case g = 5 of our result implies Hadwiger's conjecture for C 4 free graphs of suciently large chromatic number.
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).
Computing the Girth of a Planar Graph in Linear Time
, 2013
"... The girth of a graph is the minimum weight of all simple cycles of the graph. We study the problem of determining the girth of an nnode unweighted undirected planar graph. The first nontrivial algorithm for the problem, given by Djidjev, runs in O(n5/4 logn) time. Chalermsook, Fakcharoenphol, and N ..."
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The girth of a graph is the minimum weight of all simple cycles of the graph. We study the problem of determining the girth of an nnode unweighted undirected planar graph. The first nontrivial algorithm for the problem, given by Djidjev, runs in O(n5/4 logn) time. Chalermsook, Fakcharoenphol, and Nanongkai reduced the running time to O(n log2 n). Weimann and Yuster further reduced the running time to O(n logn). In this paper, we solve the problem in O(n) time.
Hadwiger Number and the Cartesian Product Operation on Graphs
, 2005
"... The Hadwiger number η(G) of a graph G is defined as the largest integer n for which the complete graph on n nodes Kn is a minor of G. Hadwiger conjectured that for any graph G, η(G) ≥ χ(G),where χ(G) is the chromatic number of G. In this paper, we investigate the Hadwiger number with respect to t ..."
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The Hadwiger number η(G) of a graph G is defined as the largest integer n for which the complete graph on n nodes Kn is a minor of G. Hadwiger conjectured that for any graph G, η(G) ≥ χ(G),where χ(G) is the chromatic number of G. In this paper, we investigate the Hadwiger number with respect to the cartesian product operation on Graphs. As the main result of this paper, we show that for any two graphs G1 and G2 with η(G1) = h and η(G2) = l, η(G1 ✷ G2) ≥ 1 4 (h − √ l) ( √ l − 2). (Since G1 ✷ G2 is isomorphic to G2 ✷ G1, we can assume without loss of generality that h ≥ l). This lower bound is the best possible (up to a small constant factor), since if G1 = Kh and G2 = Kl, η(G1 ✷ G2) ≤ 2h √ l. We also show that η(G1 ✷ G2) doesn’t have any upper bound which depends only on η(G1) and η(G2), by demonstrating graphs G1 and G2 such that η(G1) and η(G2) are bounded whereas η(G1 ✷ G2) grows with the number of nodes. (The problem of studying the Hadwiger number with respect to the cartesian product operation
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.
Hadwiger Number and the Cartesian Product Operation on Graphs
, 2006
"... The Hadwiger number is defined as the largest integer for which the complete graph on nodes . Hadwiger conjectured that for any ,where is the chromatic number of . In this thesis, we investigate the Hadwiger number with respect to the cartesian product operation on Graphs. As t ..."
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The Hadwiger number is defined as the largest integer for which the complete graph on nodes . Hadwiger conjectured that for any ,where is the chromatic number of . In this thesis, we investigate the Hadwiger number with respect to the cartesian product operation on Graphs. As the main result of this thesis, we show that for any two graphs !"# $ %&(' )*' +&,. . (Since isomorphic to / 0 , we can assume without loss of generality that 12 ). This lower bound is the best possible (up to a small constant factor), since if 3 5 6 7 !8:9 ,.' . We also show that ! are bounded whereas ;7 !8 grows with the number of nodes. (The problem of studying the Hadwiger number with respect to the cartesian product operation was posed by Z.Miller in 1978.) As consequences of our main result, we show the following: be a connected graph. Let the (unique) prime factorization of be given by !/ = C,EDGF.H3DGFIH3 KJ LM L"M is a constant. This improves the ,EDGFIH3 NJPO bound given in [2].