Let $G$ be a graph. The circumference of $G$, denoted by $cir(G)$, is the length of a longest cycle in $G$, or zero if $G$ is acyclic. In 1993, Broersma, van den Heuvel, Jung, and Veldman conjectured that, for every $t>0$, there is a constant $A=A(t)>0$ such that every 2-connected $t$-tough graph of order $n$ has circumference at least $A\log n$; the conjecture is recorded as Conjecture~2 in the 2006 survey on toughness by Bauer, Broersma, and Schmeichel. In this note, we confirm the conjecture. More precisely, every 2-connected $t$-tough graph $G$ of order $n$ satisfies $cir(G)\ge \lceil \log_k((k-1)n+1)\rceil$, where $k=\lceil 1/t\rceil+2$. The proof combines Win's bounded-degree spanning tree theorem with the theorem of Bria\'nski, Joret, Majewski, Micek, Seweryn, and Sharma that the treedepth of a 2-connected graph is at most its circumference.
For a graph $G$, let $e(G)$ denote the number of edges of $G$, and let $c(G)$ be the number of distinct cycles in $G$. Morrison, Roberts and Scott asked whether, for every fixed graph $H$ and all large $n$, some $n$-vertex $H$-free graph maximizes both $e(G)$ and $c(G)$. In this paper, we show that the answer is no for...
An $n$-vertex graph $G$ is $(\rho, d)$-dense if for all $U \subseteq V(G)$ of size at least $\rho n$, the induced subgraph $G[U]$ contains at least $\frac{d}{2}|U|^2$ edges. In 2010, partly motivated by the theory of quasirandomness, Kohayakawa, Nagle, R\"{o}dl, and Schacht conjectured that, for any fixed graph $F$, th...
A paired dominating set of a graph $G$ is a dominating set $D$ such that $G[D]$ has a perfect matching. The minimum size of such a set is the paired domination number $\gpr(G)$. Desormeaux and Henning conjectured that every cubic bipartite graph $G$ of order $n$ satisfies $\gpr(G)\le n/2$. We prove the conjecture in th...
Motivated by Mader's conjecture on connectivity keeping trees, we study the corresponding problem in triangle-free graphs. For positive integers $k,m$, we first prove that every $k$-connected triangle-free graph $G$ with $\delta(G)\geq k+\left \lceil \frac{m}{2} \right \rceil $ contains a path $P$ of order $m$ such tha...
Meng-Han Ma, Qing-Hai Liu, Li-Ping Zhang et al.· 0 citations
Let $t \ge 1$ be an integer, and let $G$ be a $t$-tough $n$-vertex graph with degree sequence $d_1, d_2, \ldots, d_n$ in non-decreasing order. In 1995, Ho\`ang conjectured that if $G$ is Hamiltonian and, for every integer $i$ satisfying $t\le i<n/2$, $d_i\le i$, and $d_{n-i+t}<n-i$, one has $d_j + d_{n-j+t} \ge n$ for...
Extending the notions of Kneser graphs and Schrijver graphs, we define the $G$-stable Kneser graph $KG(G,k)$ (where $k\geq 2$ and $G$ is a graph on $n\geq 2k$ vertices) as the graph whose vertices are the $k$-element independent sets in $G$ and whose edges join pairs of disjoint independent sets. The well-known result...
C. Arana, P. Haxell, Tomáš Kaiser et al.· 0 citations
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