Monochromatic Cycles in 2-Coloured Graphs

Benevides, Fabrício Siqueira; Łuczak, Tomasz; Scott, Alex; Skokan, Jozef; White, Matthew

doi:10.1017/s0963548312000090

Cited by 26 publications

(40 citation statements)

References 11 publications

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“…Proof of Theorem The fact that

{normalΦ}_{2} false(c false) = \frac{2}{3}

for all

c \in [\frac{3}{4}, 1)

was proved in , and the upper bound on

{normalΦ}_{2} false(c false)

was shown for all values of c in Section Let

m \geq 2

be an integer, and let

c > \frac{2 m + 1}{{false(m + 1 false)}^{2}}

. Let δ be a positive constant with

δ < min \{c - \frac{2 m + 1}{{(() m + 1)}^{2}}, \frac{1}{5 {(m + 1)}^{2}}\} .

Let G be a 2‐edge‐colored graph with sufficiently large order n and minimum degree at least

c n

.…”

Section: Proof Of Theoremmentioning

confidence: 94%

“…Benevides et al. made the following definition, allowing an asymptotic version of Question to be considered. Definition For any positive integer r and

0 < c < 1

, let

{normalΦ}_{r} false(c false)

be the supremum of the set of real‐valued ϕ such that any r ‐edge‐colored graph G of sufficiently large order n with minimum degree at least

c n

has monochromatic circumference at least

φ n

.…”

Section: The Circumferencementioning

confidence: 99%

“…In this section, edge‐colorings of graphs will be defined, which imply the upper bound on

{normalΦ}_{2} false(c false)

given in Theorem . Recall that, for

c \geq \frac{3}{4}

, it was shown in that

{normalΦ}_{2} false(c false) = \frac{2}{3}

, and so constructions will only be given for

c < \frac{3}{4}

.…”

Section: Upper Bound On φ2mentioning

confidence: 99%

“…Further, there is a 2-edge-coloring of K n such that both R and B have circumference at most 2 3 n . Theorem 1.4 implies that, as c → 1, the answer to Question 1.3 must tend to 2 3 n. Benevides et al [1] made the following definition, allowing an asymptotic version of Question 1.3 to be considered. Definition 1.5.…”

Section: The Circumferencementioning

confidence: 99%

“…Theorem 1.2 gives an exact bound on how large the circumference of a graph with a given order and minimum degree may be. This may be used to determine the value of 1 (c) for all c ∈ (0, 1). Corollary 1.6.…”

Section: The Circumferencementioning

confidence: 99%

See 4 more Smart Citations

The Monochromatic Circumference of 2‐Edge‐Colored Graphs

White¹

2016

Journal of Graph Theory

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“…Proof of Theorem The fact that

{normalΦ}_{2} false(c false) = \frac{2}{3}

for all

c \in [\frac{3}{4}, 1)

was proved in , and the upper bound on

{normalΦ}_{2} false(c false)

was shown for all values of c in Section Let

m \geq 2

be an integer, and let

c > \frac{2 m + 1}{{false(m + 1 false)}^{2}}

. Let δ be a positive constant with

δ < min \{c - \frac{2 m + 1}{{(() m + 1)}^{2}}, \frac{1}{5 {(m + 1)}^{2}}\} .

Let G be a 2‐edge‐colored graph with sufficiently large order n and minimum degree at least

c n

.…”

Section: Proof Of Theoremmentioning

confidence: 94%

“…Benevides et al. made the following definition, allowing an asymptotic version of Question to be considered. Definition For any positive integer r and