A long-standing conjecture of Thomassen says that every longest cycle of a $3$-connected graph has a chord. Thomassen (2018) proved that if $G$ is a $2$-connected cubic graph, then any longest cycle must have a chord. He also showed that in any 3-connected graph with minimum degree at least four, some longest cycle must contain a chord. Harvey proved that every longest cycle has a chord for graphs with a large minimum degree. He also conjectured that any longest cycle in a 2-connected graph with minimum degree at least three has a chord. In this paper, we prove that both Thomassen's and Harvey's conjectures are true for graphs with large circumferences. We also prove a more general result for the existence of chords in longest cycles containing a linear forest.
We consider the size of the smallest set of vertices required to intersect every longest path in a chordal graph. Such sets are known as longest path transversals. We show that if $ω(G)$ is the clique number of a chordal graph $G$, then there is a transversal of order at most $4\lceil\frac{ω(G)}{5}\rceil$. We also consider the analogous question for longest cycles, and show that if $G$ is a 2-connected chordal graph then there is a transversal intersecting all longest cycles of order at most $2\lceil\frac{ω(G)}{3}\rceil$.
Carsten Thomassen conjectured that every longest circuit in a 3-connected graph has a chord. We prove the conjecture for graphs having no $K_{3,3}$ minor, and consequently for planar graphs.
Let $G$ be a graph, $C$ a longest cycle in $G$ and $\overline{p}$, $\overline{c}$ the lengths of a longest path and a longest cycle in $G\backslash C$, respectively. Almost all lower bounds for the circumference base on a standard procedure: choose an initial cycle $C_0$ in $G$ and try to enlarge it via structures of $G\backslash C_0$ and connections between $C_0$ and $G\backslash C_0$ closely related to $\overline{p}$, $\overline{c}$ and connectivity $κ$. Actually, each lower bound obtained in result of this procedure, somehow or is related to $κ$, $\overline{p}$, $\overline{c}$ but in forms …
Let $lct(G)$ be the minimum cardinality of a set of vertices that intersects every longest cycle of a 2-connected graph $G$. We show that $lct(G)\leq k-1$ if $G$ is a partial $k$-tree and that $lct(G)\leq \max \{1, {ω(G){-}3}\}$ if $G$ is chordal, where $ω(G)$ is the cardinality of a maximum clique in $G$. Those results imply that all longest cycles intersect in 2-connected series parallel graphs and in 3-trees.
Let $G$ be a 2-connected graph of order $n$ and let $c$ be the circumference - the order of a longest cycle in $G$. In this paper we present a sharp lower bound for the circumference based on minimum degree $δ$ and $p$ - the order of a longest path in $G$. This is a common generalization of two earlier classical results for 2-connected graphs due to Dirac: (i) $c\ge \min\{n,2δ\}$; and (ii) $c\ge\sqrt{2p}$. Moreover, the result is stronger than (ii).
The circumference of a graph is the length of its longest cycles. Jackson established a conjecture of Bondy by showing that the circumference of a 3-connected cubic graph of order $n$ is $Ω(n^{0.694})$. Bilinski {\it et al.} improved this lower bound to $Ω(n^{0.753})$ by studying large Eulerian subgraphs in 3-edge-connected graphs. In this paper, we further improve this lower bound to $Ω(n^{0.8})$. This is done by considering certain 2-connected cubic graphs, finding cycles through two given edges, and distinguishing the cases whether or not these edges are adjacent.
Erdős determined the maximum size of a nonhamiltonian graph of order $n$ and minimum degree at least $k$ in 1962. Recently, Ning and Peng generalized. Erdős' work and gave the maximum size $h(n,c,k)$ of graphs with prescribed order $n$, circumference $c$ and minimum degree at least $k.$ But for some triples $n,c,k,$ the maximum size is not attained by a graph of minimum degree $k.$ For example, $h(15,14,3)=77$ is attained by a unique graph of minimum degree $7,$ not $3.$ In this paper we obtain more precise information by determining the maximum size of a graph with prescribed order, circumfer…
The circumference $c(G)$ of a graph $G$ is the length of a longest cycle. By exploiting our recent results on resistance of snarks, we construct infinite classes of cyclically $4$-, $5$- and $6$-edge-connected cubic graphs with circumference ratio $c(G)/|V(G)|$ bounded from above by $0.876$, $0.960$ and $0.990$, respectively. In contrast, the dominating cycle conjecture implies that the circumference ratio of a cyclically $4$-edge-connected cubic graph is at least $0.75$. In addition, we construct snarks with large girth and large circumference deficit, solving Problem 1 proposed in [J. Hägglu…
We show that, if a $n$-vertex triangulation $T$ of maximum degree $Δ$ has a dual that contains a cycle of length $\ell$, then $T$ has a non-crossing straight-line drawing in which some \emph{collinear set} of $Ω(\ell/Δ^4)$ vertices lie on a line. Using the current lower bounds on the length of longest cycles in 3-regular 3-connected graphs, this implies that every $n$-vertex planar graph of maximum degree $Δ$ has a collinear set of size $Ω(n^{0.8}/Δ^4)$. Very recently, Dujmović et. al. (SODA 2019) showed that, if $S$ is a collinear set in a triangulation $T$ then, for any point set $X\subset\m…
We show that every $k$-tree of toughness greater than $\frac{k}{3}$ is Hamilton-connected for $k \geq 3$. (In particular, chordal planar graphs of toughness greater than $1$ are Hamilton-connected.) This improves the result of Broersma et al. (2007) and generalizes the result of Böhme et al. (1999). On the other hand, we present graphs whose longest paths are short. Namely, we construct $1$-tough chordal planar graphs and $1$-tough planar $3$-trees, and we show that the shortness exponent of the class is $0$, at most $\log_{30}{22}$, respectively. Both improve the bound of Böhme et al. Further…
Let $G$ be a 2-connected graph, $l$ be the length of a longest path in $G$ and $c$ be the circumference - the length of a longest cycle in $G$. In 1952, Dirac proved that $c>\sqrt{2l}$ and conjectured that $c\ge 2\sqrt{l}$. In this paper we present more general sharp bounds in terms of $l$ and the length $m$ of a vine on a longest path in $G$ including Dirac's conjecture as a corollary: if $c=m+y+2$ (generally, $c\ge m+y+2$) for some integer $y\ge 0$, then $c\ge\sqrt{4l+(y+1)^2}$ if $m$ is odd; and $c\ge\sqrt{4l+(y+1)^2-1}$ if $m$ is even.
Two new sufficient conditions for generalized cycles (including Hamilton and dominating cycles as special cases) in an arbitrary k-connected graph (k=1,2,...) are derived, which prove the truth of Bondy's (1980) famous conjecture for some variants significantly improving the result expected by the given hypothesis. Similarly, two new lower bounds for the circumference (the length of a longest cycle) are established for the reverse hypothesis proposed by Jung (2001).
Let $\lpt(G)$ be the minimum cardinality of a set of vertices that intersects all longest paths in a graph $G$. Let $ω(G)$ be the size of a maximum clique in $G$, and $\tw(G)$ be the treewidth of $G$. We prove that $ \lpt(G) \leq \max\{1,ω(G)-2\}$ when $G$ is a connected chordal graph; that $\lpt(G) =1$ when $G$ is a connected bipartite permutation graph or a connected full substar graph; and that $\lpt(G) \leq \tw(G)$ for any connected graph $G$.
A Note on Grünbaum's Conjecture about Longest Cycles and Paths
Faculty of Science and Technology, Keio University（庆应义塾大学理工学部）；Faculty of Environment and Information Sciences, Yokohama National University（横滨国立大学环境信息学部）；School of Mathematics and Statistics, Key Laboratory of Nonlinear Analysis & Applications (Ministry of Education), Central China Normal University（华中师范大学数学与统计学院，非线性分析与应用教育部重点实验室）
A graph is closed when its vertices have a labeling by [n] with a certain property first discovered in the study of binomial edge ideals. In this article, we prove that a connected graph has a closed labeling if and only if it is chordal, claw-free, and has a property we call narrow, which holds when every vertex is distance at most one from all longest shortest paths of the graph.
Let $G$ be a connected graph. The eccentricity of a path $P$, denoted by ecc$_G(P)$, is the maximum distance from $P$ to any vertex in $G$. In the \textsc{Central path} (CP) problem our aim is to find a path of minimum eccentricity. This problem was introduced by Cockayne et al., in 1981, in the study of different centrality measures on graphs. They showed that CP can be solved in linear time in trees, but it is known to be NP-hard in many classes of graphs such as chordal bipartite graphs, planar 3-connected graphs, split graphs, etc. We investigate the path eccentricity of a connected graph~…
For a connected graph, the Hamiltonian cycle (path) is a simple cycle (path) that spans all the vertices in the graph. It is known from \cite{muller,garey} that HAMILTONIAN CYCLE (PATH) are NP-complete in general graphs and chordal bipartite graphs. A convex bipartite graph $G$ with bipartition $(X,Y)$ and an ordering $X=(x_1,\ldots,x_n)$, is a bipartite graph such that for each $y \in Y$, the neighborhood of $y$ in $X$ appears consecutively. $G$ is said to have convexity with respect to $X$. Further, convex bipartite graphs are a subclass of chordal bipartite graphs. In this paper, we present…
