Background Results Neighborhood Unions, Eigenvalues and Disjoint Cycles Paul Horn Department of Mathematics and Computer Science Emory University Partially based on joint work with Ron Gould, Emory University and Kazuhide Hirohata, Ibaraki National College of Technology May 13, 2011 Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
Background Results Basic Question Many questions in graph theory are of the following type: What simple structural properties of a graph imply that a graph has some more complex property? e.g. ... Dirac (1952): If G has minimum degree n 2 , then G is Hamiltonian. Ore (1960): If the degree-sum of any two non-adjacent vertices in G is at least n , then G is Hamiltonian. Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
Background Results Basic Question Many questions in graph theory are of the following type: What simple structural properties of a graph imply that a graph has some more complex property? e.g. ... Dirac (1952): If G has minimum degree n 2 , then G is Hamiltonian. Ore (1960): If the degree-sum of any two non-adjacent vertices in G is at least n , then G is Hamiltonian. Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
Background Results Cycles What if we don’t care about a Hamiltonian cycle, but only care about the existence of a cycle? Too easy: n edges imply the existence of a cycle. ... Okay, how about many cycles? Corrádi and Hajnal (1963): If G has minimum degree at least 2 k and | G | ≥ 3 k , then G contains k independent (pairwise disjoint) cycles. Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
Background Results Cycles What if we don’t care about a Hamiltonian cycle, but only care about the existence of a cycle? Too easy: n edges imply the existence of a cycle. ... Okay, how about many cycles? Corrádi and Hajnal (1963): If G has minimum degree at least 2 k and | G | ≥ 3 k , then G contains k independent (pairwise disjoint) cycles. Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
Background Results Cycles What if we don’t care about a Hamiltonian cycle, but only care about the existence of a cycle? Too easy: n edges imply the existence of a cycle. ... Okay, how about many cycles? Corrádi and Hajnal (1963): If G has minimum degree at least 2 k and | G | ≥ 3 k , then G contains k independent (pairwise disjoint) cycles. Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
Background Results Generalizations Corrádi-Hajnal: If G has minimum degree at least 2 k and | G | ≥ 3 k , then G contains k independent cycles. Special case: If | G | is divisible by 3, and G has minimum degree at least 2 3 n , then G has a triangle-factor. A generalization? Hajnal-Szeméredi (1970): If | G | is divisible by k , and G has minimum degree at least k − 1 k n , then G has a K k -factor. Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
Background Results Generalizations Corrádi-Hajnal: If G has minimum degree at least 2 k and | G | ≥ 3 k , then G contains k independent cycles. Special case: If | G | is divisible by 3, and G has minimum degree at least 2 3 n , then G has a triangle-factor. A generalization? Hajnal-Szeméredi (1970): If | G | is divisible by k , and G has minimum degree at least k − 1 k n , then G has a K k -factor. Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
Background Results Generalizations Corrádi-Hajnal: If G has minimum degree at least 2 k and | G | ≥ 3 k , then G contains k independent cycles. Special case: If | G | is divisible by 3, and G has minimum degree at least 2 3 n , then G has a triangle-factor. A generalization? Hajnal-Szeméredi (1970): If | G | is divisible by k , and G has minimum degree at least k − 1 k n , then G has a K k -factor. Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
Background Results Generalizations Corrádi-Hajnal: If G has minimum degree at least 2 k and | G | ≥ 3 k , then G contains k independent cycles. Different degree conditions? Wang (1999): If the degree sum of any two non-adjacent vertices is at least 4 k − 1, and | G | ≥ 3 k , then G contains k independent cycles. Remarks: Ore-type condition (compared to Dirac-type condition on Corrádi-Hajnal) Strengthens results of Justensen (1989) Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
Background Results Degree conditions Dirac-type conditions: Ore-type conditions: Minimum degree condition: Degree-Sum condition: Every vertex must have some Vertices may have small degree, minimum degree degree sum of non-adjacent vertices must be large. Both conditions: Guarantee some ’local expansion’. Dirac: Requires it at every vertex. Ore: Requires it at ’most vertices’. Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
Background Results Degree conditions Dirac-type conditions: Ore-type conditions: Minimum degree condition: Degree-Sum condition: Every vertex must have some Vertices may have small degree, minimum degree degree sum of non-adjacent vertices must be large. Both conditions: Guarantee some ’local expansion’. Dirac: Requires it at every vertex. Ore: Requires it at ’most vertices’. Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
Background Results Degree conditions Dirac-type conditions: Ore-type conditions: Minimum degree condition: Degree-Sum condition: Every vertex must have some Vertices may have small degree, minimum degree degree sum of non-adjacent vertices must be large. Split the difference! Neighborhood union condition: Require than the union of the neighborhood of two vertices is large. Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
Background Results Degree conditions Dirac-type conditions: Ore-type conditions: Minimum degree condition: Degree-Sum condition: Every vertex must have some Vertices may have small degree, minimum degree degree sum of non-adjacent vertices must be large. Neighborhood union condition: Require than the union of the neighborhood of two vertices is large. Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
Background Results Corrádi-Hajnal Revisited Corrádi-Hajnal: If G has minimum degree at least 2 k and | G | ≥ 3 k , then G contains k independent cycles. Gould and J. Faudree (2005): If the neighborhood union of any two non-adjacent vertices in G has size at least 3 k , then G contains k independent cycles. Remark: Condition implies | G | ≥ 3 k . Best possible? Gould and Faudree conjectured union of 2 k + O ( 1 ) might suffice. (Union of 2 k not sufficient.) Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
Background Results Corrádi-Hajnal Revisited Corrádi-Hajnal: If G has minimum degree at least 2 k and | G | ≥ 3 k , then G contains k independent cycles. Gould and J. Faudree (2005): If the neighborhood union of any two non-adjacent vertices in G has size at least 3 k , then G contains k independent cycles. Remark: Condition implies | G | ≥ 3 k . Best possible? Gould and Faudree conjectured union of 2 k + O ( 1 ) might suffice. (Union of 2 k not sufficient.) Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
Background Results Corrádi-Hajnal Revisited Corrádi-Hajnal: If G has minimum degree at least 2 k and | G | ≥ 3 k , then G contains k independent cycles. Theorem (Gould, Hirohata, H.) If the neighborhood union of any two non-adjacent vertices in G has size at least 2 k + 1 , and | G | ≥ 30 k then G contains k independent cycles. Remark: 2 k + 1 is best possible, but | G | > 30 k is not. Improvement to | G | ≥ 3 k is impossible, however: consider an isolated vertex and a K 3 k − 1 . Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
Background Results Proof idea: As in the proofs of many similar results, we proceed inductively: Start with a system of k − 1 cycles, C 1 , . . . , C k − 1 with � | C k | minimized. Remainder is a forest (or we’re done!) Use minimality + degree conditions to imply forest is ’nice’ Use ’nice’ forest and cycles, to find another cycle. Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
Background Results Proof idea: As in the proofs of many similar results, we proceed inductively: Start with a system of k − 1 cycles, C 1 , . . . , C k − 1 with � | C k | minimized. Remainder is a forest (or we’re done!) Use minimality + degree conditions to imply forest is ’nice’ Use ’nice’ forest and cycles, to find another cycle. Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
Background Results Proof idea: As in the proofs of many similar results, we proceed inductively: Start with a system of k − 1 cycles, C 1 , . . . , C k − 1 with � | C k | minimized. Remainder is a forest (or we’re done!) Use minimality + degree conditions to imply forest is ’nice’ Use ’nice’ forest and cycles, to find another cycle. Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
Background Results Proof idea: As in the proofs of many similar results, we proceed inductively: Start with a system of k − 1 cycles, C 1 , . . . , C k − 1 with � | C k | minimized. Remainder is a forest (or we’re done!) Use minimality + degree conditions to imply forest is ’nice’ Use ’nice’ forest and cycles, to find another cycle. Horn Neighborhood Unions, Eigenvalues, and Disjoint Cycles
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