Panconnected graphs II

Denote by R(L, L, L) the minimum integer N such that any 3-coloring of the edges of the complete graph K N contains a monochromatic copy of a graph L. Bondy and Erdős conjectured that for an odd cycle on n vertices C n , This is sharp if true.Luczak proved that if n is odd, then R(C n , C n , C n ) = 4n + o(n), as n → ∞. We prove here the exact Bondy-Erdős conjecture for sufficiently large values of n. We also describe the Ramsey-extremal colorings and prove some related stability theorems.

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“…Lemma 10 (Williamson, [25]). Suppose that G N is a t-complete graph on N vertices for some integer 1 ≤ t ≤ N/2 − 2.…”

Section: Proof Of Theorem 5 21 Density Statementsmentioning

confidence: 99%

The 3-colored Ramsey number of odd cycles

Kohayakawa¹,

Simonovits²,

Skokan³

2005

Electronic Notes in Discrete Mathematics

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“…The bipartite analogue of Williamson's panconnectivity theorem in [8] was proved recently by Hui et al…”

Section: Panconnectivity and Circumference Of Bigraphsmentioning

confidence: 92%

Locating Pairs of Vertices on a Hamiltonian Cycle in Bigraphs

Faudree

Lehel

Yoshimoto

2015

Graphs and Combinatorics

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“…, v l are the vertices of T 1 in the path from x to y in H s 1 . By Claim 2.3, (G s ) (|G s | + 2)/2 and hence G s is panconnected [9], that is, there is a path of length i between any two vertices in G s , for all 2 i < |G s |. By Claim 2.5, |G s | > t 1 + 1 l + 1 and hence there is a path p, u 1 , u 2 , .…”

Section: Proofmentioning

confidence: 93%

Subdivisions of graphs: A generalization of paths and cycles

Babu

Diwan

2008

Discrete Mathematics

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One of the basic results in graph theory is Dirac's theorem, that every graph of order n 3 and minimum degree n/2 is Hamiltonian. This may be restated as: if a graph of order n and minimum degree n/2 contains a cycle C then it contains a spanning cycle, which is just a spanning subdivision of C. We show that the same conclusion is true if instead of C, we choose any graph H such that every connected component of H is non-trivial and contains at most one cycle. The degree bound can be improved to (n − t)/2 if H has t components that are trees.We attempt a similar generalization of the Corrádi-Hajnal theorem that every graph of order 3k and minimum degree 2k contains k disjoint cycles. Again, this may be restated as: every graph of order 3k and minimum degree 2k contains a subdivision of kK 3 . We show that if H is any graph of order n with k components, each of which is a cycle or a non-trivial tree, then every graph of order n and minimum degree n − k contains a subdivision of H.

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Panconnected graphs II

Cited by 47 publications

References 11 publications

The 3-colored Ramsey number of odd cycles

The 3-colored Ramsey number of odd cycles

Locating Pairs of Vertices on a Hamiltonian Cycle in Bigraphs

Subdivisions of graphs: A generalization of paths and cycles

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