Circumference of a graph and its distance dominating longest cycles

Abstract

In this note, we prove the following: Let $G$ be a $k$-connected graph ($k\ge 2$) with circumference $c\left(G\right)$ and $m$ a non-negative integer. Then

• [(1)] Either $c\left(G\right)\ge (2m+2)k$, or $d_G(v,C)\le m$ for any longest cycle $C$ and any vertex $v$ of $G$.

• [(2)] Either $c\left(G\right)\ge (2m+3)k$, or $d_G(e,C)\le m$ for any longest cycle $C$ and any edge $e$ of $G$.

When $m=0$, $C$ in (1) and (2) are well-known Hamiltonian cycle and dominating longest cycle, respectively. Moreover, we give graphs to show that the bounds on $c\left(G\right)$ are all sharp, even for those graphs that are triangle-free with only possible exception $m=0$ in (2). (1) is also best possible for those graphs that are bipartite.

Introduction

For graph-theoretic notation not explained in this note, we refer the reader to [3]. We consider only simple graphs in this note. The circumference of a graph $G$ is the length of a longest cycle, denoted by $c\left(G\right)$. We use $n\left(G\right)$, $\delta \left(G\right)$ and $\kappa \left(G\right)$ to denote the order, minimum degree and connectivity of a graph $G$. By $m$, we always mean a non-negative integer.

The distance $d_G(H_1,H_2)$ between two subgraphs $H_1$ and $H_2$ of $G$ is defined to be $min\{d_G(v_1,v_2):v_1\in V\left(H_1\right),v_2\in V\left(H_2\right)\}$, where $d_G(v_1,v_2)$ denotes the length of a shortest path between $v_1$ and $v_2$ in $G$.

A cycle $C$ is $m$-dominating if for all $x\in V\left(G\right)$, $d_G(x,C)\le m$, that is, all the vertices of $V(G-C)$ are at distance at most $m$ from $C$. Clearly, every 0-dominating cycle is Hamiltonian. A cycle $C$ is $m$-edge-dominating if for all $e\in E\left(G\right)$, $d_G(e,C)\le m$. Clearly, a cycle is 0-edge-dominating (or simply dominating) if every edge of $G$ is incident with a vertex of $C$, $G-V\left(C\right)$ is edgeless.

An $m$-dominating cycle is also $m$-edge-dominating because both terminal vertices of any edge are at distance at most $m$ from the $m$-dominating cycle, but the reverse is not true in general. An $m$-edge-dominating cycle is only $(m+1)$-dominating.

It is very popular to decide whether there is a Hamiltonian cycle (0-dominating cycle) in a graph. There are also many results on whether a longest cycle is (0-edge-) dominating.

In 1971, Nash-Williams proved the first fundamental result concerning dominating cycles.

Theorem 1 Nash-Williams [10]

Let $G$ be a 2-connected graph. If $\delta \left(G\right)\ge (n\left(G\right)+2)∕3$, then every longest cycle of $G$ is dominating.

The reverse version of this theorem was established by Voss and Zuluaga.

Theorem 2 Voss and Zuluaga [13]

Let $G$ be a 3-connected graph with $c\left(G\right)\le 3\delta \left(G\right)-4$, then every longest cycle of $G$ is dominating.

The bounds in Theorem 1, Theorem 2 can be essentially improved by incorporating connectivity $\kappa \left(G\right)$ into these bounds.

Theorem 3 Yamashita [15]

Let $G$ be a 3-connected graph. If $\delta \left(G\right)\ge (n\left(G\right)+\kappa \left(G\right)+3)∕4$, then every longest cycle of $G$ is dominating.

Theorem 4 Nikoghosyan and Nikoghosyan [11]

Let $G$ be a 4-connected graph with $c\left(G\right)\le 4\delta \left(G\right)-\kappa \left(G\right)-5$, then every longest cycle of $G$ is dominating.

More results on dominating cycles can be found in [5], [9], [12]. The above results are all on 0-edge-dominating longest cycles. A natural problem is how to extend these results to any general $m$-dominating longest cycles or $m$-edge-dominating longest cycles.

Broersma and Fraisse independently considered $m$-dominating cycles and proved a result that is a little stronger than the conjecture proposed by Bondy and Fan [2], in which the case when $m=0$, it is well-known as Chvátal–Erdős theorem [6].

Theorem 5 Broersma [4], Fraisse [8]

Let $G$ be a $k$-connected graph with no set of cardinality $k+1$, whose vertices are pairwise at distance at least $2m+2$. Then $G$ has an $m$-dominating cycle.

Note that any $k$ vertices in a $k$-connected graph lie on a common cycle. If a $k$-connected graph $G$ has no $m$-dominating cycle, by Theorem 5, any $k$ vertices of a set whose vertices are at distance at least $2m+2$ lie on a common cycle. Such cycle is of length at least $(2m+2)k$. Hence, Corollary 6 follows from Theorem 5.

Corollary 6

Let $G$ be a graph with $\kappa \left(G\right)=k⩾2$. If $c\left(G\right)\le (2m+2)k-1$, then $G$ has an $m$-dominating cycle.

We firstly prove a result that is a little stronger than Corollary 6, which cannot be deprived from Theorem 5.

Theorem 7

Let $G$ be a graph with $\kappa \left(G\right)=k⩾2$. If $c\left(G\right)\le (2m+2)k-1$, then every longest cycle of $G$ is $m$-dominating.

Furthermore, we consider $m$-edge-dominating cycles and obtain the following result.

Theorem 8

Let $G$ be a graph with $\kappa \left(G\right)=k⩾2$. If $c\left(G\right)\le (2m+3)k-1$, then every longest cycle of $G$ is $m$-edge-dominating.

An immediate consequence of the theorem can be obtained.

Corollary 9

Let $G$ be a graph with $\kappa \left(G\right)=k⩾2$. If $c\left(G\right)\le (2m+3)k-1$, then $G$ has an $m$-edge-dominating cycle.

The bounds on $c\left(G\right)$ in Theorems 7, 8 and Corollaries 6, 9 are all sharp, which will be shown in Section 3. Proofs of main results are presented in Section 2. The last section is devoted to the concluding remarks.