Abstract
The existence of cycles with a given length is classical topic in graph theory with a plethora of open problems. Examples related to the main result of this paper include a conjecture of Burr and Erdős from 1976 asked whether for every integer $m$ and a positive odd integer $k$, there exists $d$ such that every graph with average degree at least $d$ contains a cycle of length $m$ modulo $k$; this conjecture was proven by Bollobás in [Bull. London Math. Soc. 9 (1977), 97-98]( https://doi.org/10.1112/blms/9.1.97). Another example is a problem of Erdős from the 1990s asking whether there exists $A\subseteq\mathbb{N}$ with zero density and constants $n_0$ and $d_0$ such that every graph with at least $n_0$ vertices and the average degree at least $d_0$ contains a cycle with length in the set $A$, which was resolved by Verstraete in [J. Graph Theory 49 (2005), 151-167]( https://doi.org/10.1002/jgt.20072). In 1983, Thomassen conjectured that for all integers $m$ and $k$, every graph with minimum degree $k+1$ contains a cycle of length $2m$ modulo $k$. Note that the parity condition in the first and the third conjectures is necessary because of bipartite graphs. The current paper contributes to this long line of research by proving that for every integer $m$ and a positive odd integer $k$, every sufficiently large $3$-connected cubic graph contains a cycle of length $m$ modulo $k$. The result is the best possible in the sense that the same conclusion is not true for $2$-connected cubic graphs or $3$-connected graphs with minimum degree three.
Highlights
1.1 Cycle lengths modulo kLet G be a graph and let A be a set of natural numbers
Note that the restriction to odd natural numbers k is necessary since bipartite graphs contain no cycles of odd length
For every odd natural number k, there exists a natural number N(k) such that every 3-connected cubic graph with at least N(k) vertices contains a cycle of length m modulo k for every natural number m
Summary
Let G be a graph and let A be a set of natural numbers. Which properties of G and A guarantee the existence of a cycle in G whose length is in A? Erdos asked many years ago whether there exists a set A ⊂ N of density zero and two constants cA, nA such that every graph with at least nA vertices and average degree at least cA contains a cycle whose length is in A, see [5]. In 1976, Erdos and Burr [8] conjectured that for all natural numbers m and k where k is odd, there exists a constant ck(m) such that every graph with average degree at least ck(m) has a cycle of length m modulo k. Thomassen [18] proved that a graph with minimum degree at least 3 and girth at least 2(k2 + 1)(3 · 2k2+1 + (k2 + 1)2 − 1) contains cycles of all even lengths modulo k, which is the strongest known result for the class of cubic graphs. K, and N such that k ≥ 12 and m and k are divisible by 3, we construct a 2-connected cubic graph on at least N vertices which has no cycles of length m modulo k
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