SOLVED

If $G$ is a graph with infinite chromatic number and $a_1<a_2<\cdots $ are lengths of the odd cycles of $G$ then $\sum \frac{1}{a_i}=\infty$.

SOLVED

If $G$ is a graph which contains odd cycles of $\leq k$ different lengths then $\chi(G)\leq 2k+2$, with equality if and only if $G$ contains $K_{2k+2}$.

Conjectured by Bollobás and Erdős. Bollobás and Shelah have confirmed this for $k=1$. Proved by Gyárfás [Gy92], who proved the stronger result that, if $G$ is 2-connected, then $G$ is either $K_{2k+2}$ or contains a vertex of degree at most $2k$.

A stronger form was established by Gao, Huo, and Ma [GaHuMa21], who proved that if a graph $G$ has chromatic number $\chi(G)\geq 2k+3$ then $G$ contains cycles of $k+1$ consecutive odd lengths.

OPEN

Does every graph on $n$ vertices with $>\mathrm{ex}(n;C_4)$ edges contain $\gg n^{1/2}$ many copies of $C_4$?

Conjectured by Erdős and Simonovits, who could not even prove that at least $2$ copies of $C_4$ are guaranteed.

He, Ma, and Yang [HeMaYa21] have proved this conjecture when $n=q^2+q+1$ for some even integer $q$.

OPEN - $1000

Does every graph with minimum degree at least 3 contain a cycle of length $2^k$ for some $k\geq 2$?

Conjectured by Erdős and Gyárfás, who believed the answer must be negative, and in fact for every $r$ there must be a graph of minimum degree at least $r$ without a cycle of length $2^k$ for any $k\geq 2$.

This was solved in the affirmative if the minimum degree is larger than some absolute constant by Liu and Montgomery [LiMo20] (therefore disproving the above stronger conjecture of Erdős and Gyárfás). Liu and Montgomery prove a much stronger result: if the average degree of $G$ is sufficiently large then there is some large integer $\ell$ such that for every even integer $m\in [(\log \ell)^8,\ell]$, $G$ contains a cycle of length $m$.

OPEN

Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_1<a_2<\cdots $ be the lengths of cycles in $G$. Is it true that
\[\sum\frac{1}{a_i}\gg \log k?\]
Is the sum $\sum\frac{1}{a_i}$ minimised when $G$ is a complete bipartite graph?

A problem of Erdős and Hajnal. Gyárfás, Komlós, and Szemerédi [GyKoSz84] have proved that this sum is $\gg \log k$. Liu and Montgomery [LiMo20] have proved the asymptotically sharp lower bound of $\geq (\tfrac{1}{2}-o(1))\log k$.

See also the entry in the graphs problem collection.

See also [57].

SOLVED

Is it true that for every infinite arithmetic progression $P$ which contains even numbers there is some constant $c=c(P)$ such that every graph with average degree at least $c$ contains a cycle whose length is in $P$?

SOLVED - $100

Is there a set $A\subset \mathbb{N}$ of density $0$ and a constant $c>0$ such that every graph on sufficiently many vertices with average degree $\geq c$ contains a cycle whose length is in $A$?

Bollobás [Bo77] proved that such a $c$ does exist if $A$ is an infinite arithmetic progression containing even numbers (see [71]).

Erdős was 'almost certain' that if $A$ is the set of powers of $2$ then no such $c$ exists (although he conjectured that $n$ vertices and average degree $\gg (\log n)^{C}$ suffices for some $C=O(1)$). If $A$ is the set of squares (or the set of $p\pm 1$ for $p$ prime) then he had no guess.

Solved by Verstraëte [Ve05], who gave a non-constructive proof that such a set $A$ exists.

Liu and Montgomery [LiMo20] proved that in fact this is true when $A$ is the set of powers of $2$ (more generally any set of even numbers which doesn't grow too quickly) - in particular this contradicts the previous belief of Erdős.

OPEN - $500

Let $f(n)\to \infty$ (possibly very slowly). Is there a graph of infinite chromatic number such that every finite subgraph on $n$ vertices can be made bipartite by deleting at most $f(n)$ edges?

Conjectured by Erdős, Hajnal, and Szemerédi [EHS82].

Rödl [Ro82] has proved this for hypergraphs, and also proved there is such a graph (with chromatic number $\aleph_0$) if $f(n)=\epsilon n$ for any fixed constant $\epsilon>0$.

It is open even for $f(n)=\sqrt{n}$. Erdős offered \$500 for a proof but only \$250 for a counterexample. This fails (even with $f(n)\gg n$) if the graph has chromatic number $\aleph_1$ (see [111]).

OPEN

The cycle set of a graph $G$ on $n$ vertices is a set $A\subseteq \{3,\ldots,n\}$ such that there is a cycle in $G$ of length $\ell$ if and only if $\ell \in A$. Let $f(n)$ count the number of possible such $A$.

Prove that $f(n)=o(2^n)$.

Prove that $f(n)/2^{n/2}\to \infty$.

Conjectured by Erdős and Faudree, who showed that $2^{n/2}<f(n) \leq 2^{n-2}$. The first problem was solved by Verstraëte [Ve04], who proved
\[f(n)\ll 2^{n-n^c}\]
for some constant $c>0$.

One can also ask about the existence and value of $\lim f(n)^{1/n}$.

OPEN

For every $r\geq 4$ and $k\geq 2$ is there some finite $f(k,r)$ such that every graph of chromatic number $\geq f(k,r)$ contains a subgraph of girth $\geq r$ and chromatic number $\geq k$?

Conjectured by Erdős and Hajnal. Rödl [Ro77] has proved the $r=4$ case. The infinite version (whether every graph of infinite chromatic number contains a subgraph of infinite chromatic number whose girth is $>k$) is also open.

In [Er79b] Erdős also asks whether \[\lim_{k\to \infty}\frac{f(k,r+1)}{f(k,r)}=\infty.\]

OPEN

Is there some $F(n)$ such that every graph with chromatic number $\aleph_1$ has, for all large $n$, a subgraph with chromatic number $n$ on at most $F(n)$ vertices?

Conjectured by Erdős, Hajnal, and Szemerédi [EHS82]. This fails if the graph has chromatic number $\aleph_0$.

OPEN

Any graph on $n$ vertices can be decomposed into $O(n)$ many cycles and edges.

OPEN

Let $G$ be a graph with $n$ vertices and $\delta n^{2}$ edges. Are there subgraphs $H_1,H_2\subseteq G$ such that

- $H_1$ has $\gg \delta^3n^2$ edges and every two edges in $H_1$ are contained in a cycle of length at most $6$, and furthermore if two edges share a vertex they are on a cycle of length $4$, and
- $H_2$ has $\gg \delta^2n^2$ edges and every two edges in $H_2$ are contained in a cycle of length at most $8$.

A problem of Erdős, Duke, and Rödl. Duke and Erdős [DuEr83], who proved the first if $n$ is sufficiently large depending on $\delta$. The real challenge is to prove this when $\delta=n^{-c}$ for some $c>0$. Duke, Erdős, and Rödl [DER84] proved the first statement with a $\delta^5$ in place of a $\delta^3$.

Fox and Sudakov [FoSu08b] have proved the second statement when $\delta >n^{-1/5}$.

OPEN

What is the maximum number of edges that a graph on $n$ vertices can have if it does not contain two edge-disjoint cycles with the same vertex set?

Pyber, Rödl, and Szemerédi [PRS95] constructed such a graph with $\gg n\log\log n$ edges.

Chakraborti, Janzer, Methuku, and Montgomery [CJMM24] have shown that such a graph can have at most $n(\log n)^{O(1)}$ many edges. Indeed, they prove that there exists a constant $C>0$ such that for any $k\geq 2$ there is a $c_k$ such that if a graph has $n$ vertices and at least $c_kn(\log n)^{C}$ many edges then it contains $k$ pairwise edge-disjoint cycles with the same vertex set.

OPEN

Let $k\geq 4$ and $g_k(n)$ denote the largest $m$ such that there is a graph on $n$ vertices with chromatic number $k$ and girth $m$. Does
\[\lim_{n\to \infty}\frac{g_k(n)}{\log n}\]
exist?