On graphs without cycles of length 1 modulo 3
Yandong Bai, Binlong Li, Yufeng Pan, Shenggui Zhang
TL;DR
The paper determines the exact extremal number for graphs with no cycles of length 1 modulo 3. It proves that for n vertices, the maximum number of edges is governed by a partition n−1=9q+r (0≤r≤8), giving an upper bound of 15q+⌊3r/2⌋ and showing equality when the graph is built from Petersen blocks, which also yields c_{1,3}=5/3. The methodology combines detailed cycle-structure analysis, a no-two-disjoint-cycles condition, and an induction on n, culminating in a complete description of extremal graphs. These results complete the classification for k≤4 and emphasize Petersen graphs as the fundamental extremal components in this modular-cycle setting.
Abstract
Burr and Erdős conjectured in 1976 that for every two integers $k>\ell\geqslant 0$ satisfying that $k\mathbb{Z}+\ell$ contains an even integer, an $n$-vertex graph containing no cycles of length $\ell$ modulo $k$ can contain at most a linear number of edges on $n$. Bollobás confirmed this conjecture in 1977 and then Erdős proposed the problem of determining the exact value of the maximum number of edges in such a graph. For the above $k$ and $\ell$, define $c_{\ell,k}$ to be the least constant such that every $n$-vertex graph with at least $c_{\ell,k}\cdot n$ edges contains a cycle of length $\ell$ modulo $k$. The precise (or asymptotic) values of $c_{\ell,k}$ are known for very few pairs $\ell$ and $k$. In this paper, we precisely determine the maximum number of edges in a graph containing no cycles of length 1 modulo 3. In particular, we show that every $n$-vertex graph with at least $\frac{5}{3}(n-1)$ edges contains a cycle of length 1 modulo 3, unless $9|(n-1)$ and each block of the graph is a Petersen graph. As a corollary, we obtain that $c_{1,3}=\frac{5}{3}$. This is the last remaining class modulo $k$ for $1\leqslant k\leqslant 4$.