The Erd?s–Gyárfás function with respect to Gallai?colorings
نویسندگان
چکیده
For fixed $p$ and $q$, an edge-coloring of the complete graph $K_n$ is said to be a $(p, q)$-coloring if every $K_p$ receives at least $q$ distinct colors. The function $f(n, p, q)$ minimum number colors needed for have q)$-coloring. This was introduced about 45 years ago, but studied systematically by Erd\H{o}s Gy\'{a}rf\'{a}s in 1997, now known as Erd\H{o}s-Gy\'{a}rf\'{a}s function. In this paper, we study with respect Gallai-colorings, where Gallai-coloring without rainbow triangles. Combining two concepts, consider $g(n, that Gallai-$(p, $K_n$. Using anti-Ramsey $K_3$, nontrivial only $2\leq q\leq p-1$. We give general lower bound how falls off from being equal $n-1$ when $q=p-1$ $p\geq 4$ $\Theta(\log n)$ $q = 2$. particular, appropriate $n$, prove $g=n-c$ $q=p-c$ $c\in \{1,2\}$, $g$ most fractional power $n$ $q=\lfloor\sqrt{p-1}\rfloor$, logarithmic \lfloor\log_2 (p-1)\rfloor+1$.
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ژورنال
عنوان ژورنال: Journal of Graph Theory
سال: 2022
ISSN: ['0364-9024', '1097-0118']
DOI: https://doi.org/10.1002/jgt.22822