Monochromatic odd cycles in edge-coloured complete graphs
It is easy to see that every $q$-edge-colouring of the complete graph on $2^q+1$ vertices must contain a monochromatic odd cycle. A natural question raised by Erd\H{o}s and Graham in $1973$ asks for the smallest $L(q)$ such that every $q$-edge-colouring of $K_{2^q+1}$ must contain a monochromatic od...
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| creator | Girão, António Hunter, Zach |
| description | It is easy to see that every $q$-edge-colouring of the complete graph on
$2^q+1$ vertices must contain a monochromatic odd cycle. A natural question
raised by Erd\H{o}s and Graham in $1973$ asks for the smallest $L(q)$ such that
every $q$-edge-colouring of $K_{2^q+1}$ must contain a monochromatic odd cycle
of length at most $L(q)$. In here, we show that
$L(q)=O\left(\frac{2^q}{q^{1-o(1)}}\right)$ giving the first non-trivial upper
bound on $L(q)$. |
| doi_str_mv | 10.48550/arxiv.2412.07708 |
| format | Article |
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$2^q+1$ vertices must contain a monochromatic odd cycle. A natural question
raised by Erd\H{o}s and Graham in $1973$ asks for the smallest $L(q)$ such that
every $q$-edge-colouring of $K_{2^q+1}$ must contain a monochromatic odd cycle
of length at most $L(q)$. In here, we show that
$L(q)=O\left(\frac{2^q}{q^{1-o(1)}}\right)$ giving the first non-trivial upper
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$2^q+1$ vertices must contain a monochromatic odd cycle. A natural question
raised by Erd\H{o}s and Graham in $1973$ asks for the smallest $L(q)$ such that
every $q$-edge-colouring of $K_{2^q+1}$ must contain a monochromatic odd cycle
of length at most $L(q)$. In here, we show that
$L(q)=O\left(\frac{2^q}{q^{1-o(1)}}\right)$ giving the first non-trivial upper
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$2^q+1$ vertices must contain a monochromatic odd cycle. A natural question
raised by Erd\H{o}s and Graham in $1973$ asks for the smallest $L(q)$ such that
every $q$-edge-colouring of $K_{2^q+1}$ must contain a monochromatic odd cycle
of length at most $L(q)$. In here, we show that
$L(q)=O\left(\frac{2^q}{q^{1-o(1)}}\right)$ giving the first non-trivial upper
bound on $L(q)$.</abstract><doi>10.48550/arxiv.2412.07708</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Combinatorics |
| title | Monochromatic odd cycles in edge-coloured complete graphs |
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