The Monochromatic Circumference of 2‐Edge‐Colored Graphs

White, Matthew

doi:10.1002/jgt.22052

Cited by 6 publications

(8 citation statements)

References 13 publications

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“…, and let G be a graph on n ≥ n 0 vertices with δ(G) ≥ (3/4 + γ)n. Apply Lemma 5.1 to G to get a balanced bipartite (ǫ, d)-reduced graph Γ on 2k vertices with minimum degree at least (3/4 + γ/2)k and then apply Theorem 4.2 to Γ. If Γ has a monochromatic connected matching of size at least (1/2 + 2η)k, then apply Lemma 5.4 to get a cycle of length at least (1 + η)n. Otherwise, Γ is 4η-extremal, so by Lemma 7.1, G is 8η-extremal and since 16 √ 8η ≤ 64 √ η ≤ γ we may apply Proposition 8.5 to G to finish the proof.…”

Section: Stabilitymentioning

confidence: 98%

“…Finally we state the lemma which allows us to turn the connected matching in the reduced graph into the cycle in the original graph. Some variant of this lemma, first introduced by Luczak [13], has been utilized by many authors, in particular [8], [2], and [16]. See Lemma 2.2 in [2] for the variant of Luczak's lemma which is used to build the nearly spanning paths in each pair (in place of the much stronger blow-up lemma).…”

Section: Regularity: From Connected Matchings To Cyclesmentioning

confidence: 99%

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Long monochromatic paths and cycles in 2-colored bipartite graphs

DeBiasio

Krueger

2020

Discrete Mathematics

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Gyárfás and Lehel and independently Faudree and Schelp proved that in any 2coloring of the edges of K n,n there exists a monochromatic path on at least 2⌈n/2⌉ vertices, and this is tight. We prove a stability version of this result which holds even if the host graph is not complete; that is, if G is a balanced bipartite graph on 2n vertices with minimum degree at least (3/4 + o(1))n, then in every 2-coloring of the edges of G, either there exists a monochromatic cycle on at least (1 + o(1))n vertices, or the coloring of G is close to an extremal coloring -in which case G has a monochromatic path on at least 2⌈n/2⌉ vertices and a monochromatic cycle on at least 2⌊n/2⌋ vertices. Furthermore, we determine an asymptotically tight bound on the length of a longest monochromatic cycle in a 2-colored balanced bipartite graph on 2n vertices with minimum degree δn for all 0 ≤ δ ≤ 1.

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Section: Stabilitymentioning

confidence: 98%

Section: Regularity: From Connected Matchings To Cyclesmentioning

confidence: 99%

Long monochromatic paths and cycles in 2-colored bipartite graphs

DeBiasio

Krueger

2020

Discrete Mathematics

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“…(White [14]) Let G be a graph of order n such that for some integer m 3, δ(G) 2m−1 m 2 n. If the edges of G are 2-colored then there is a monochromatic component of order at least n m−1 . This result is basically implicit in the proof of Lemma 4.7 in White [14] (see also in [15]); however, it is not even stated there as a separate statement. Note that Theorem 4.1 is false for m = 2, in which case Lemma 1.1 gives the order of the largest monochromatic component.…”

mentioning

confidence: 99%

Large Monochromatic Components in Edge Colored Graphs with a Minimum Degree Condition

Gyárfás

Sárközy

2017

Electron. J. Combin.

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It is well-known that in every $k$-coloring of the edges of the complete graph $K_n$ there is a monochromatic connected component of order at least ${n\over k-1}$. In this paper we study an extension of this problem by replacing complete graphs by graphs of large minimum degree. For $k=2$ the authors proved that $\delta(G)\ge{3n\over 4}$ ensures a monochromatic connected component with at least $\delta(G)+1$ vertices in every $2$-coloring of the edges of a graph $G$ with $n$ vertices. This result is sharp, thus for $k=2$ we really need a complete graph to guarantee that one of the colors has a monochromatic connected spanning subgraph. Our main result here is that for larger values of $k$ the situation is different, graphs of minimum degree $(1-\epsilon_k)n$ can replace complete graphs and still there is a monochromatic connected component of order at least ${n\over k-1}$, in fact $$\delta(G)\ge \left(1 - \frac{1}{1000(k-1)^9}\right)n$$ suffices.Our second result is an improvement of this bound for $k=3$. If the edges of $G$ with $\delta(G)\geq {9n\over 10}$ are $3$-colored, then there is a monochromatic component of order at least ${n\over 2}$. We conjecture that this can be improved to ${7n\over 9}$ and for general $k$ we conjecture the following: if $k\geq 3$ and $G$ is a graph of order $n$ such that $\delta(G)\geq \left( 1 - \frac{k-1}{k^2}\right)n$, then in any $k$-coloring of the edges of $G$ there is a monochromatic connected component of order at least ${n\over k-1}$.

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“…|A 123 | > N/4.Proof. For A 12 = ∅ the assertion follows from(14). If A 12 = ∅, then, by (15) and Claim 26, we get|A 123 | = N − |A 1 | − |A 2 | − |A 3 | − |A 12 | > N/4 .Let us also note the following consequence of Claim 26.Claim 28.…”

mentioning

confidence: 83%

“…Schelp [12] observed that for sparse graphs H, such as paths or cycles, we may expect that G → (H) r for all G which have R r (H) vertices, provided the minimum degree of G is large enough. For H which is a path or a cycle and just two colours this problem has been thoroughly studied in a series of papers (see [1,6,9,13,14]), so now we know the minimum value of the constant c for which, in the relation K N → (C n ) 2 , the complete graphs K N can be replaced by G with δ(G) ≥ cN , at least for large n. Not surprisingly, the optimal constant c depends on the parity of n.…”

Section: Introductionmentioning

confidence: 99%