Characterizing forbidden subgraphs that imply pancyclicity in 4-connected, claw-free graphs

Abstract

In 1984, Matthews and Sumner conjectured that every 4-connected, claw-free graph contains a Hamiltonian cycle. This still unresolved conjecture has been the motivation for research into the existence of other cycle structures. In this paper, we consider the stronger property of pancyclicity for 4-connected graphs. In particular, we show that every 4-connected, $\{K_{1,3},N(i,j,k)\}$-free graph, where $i,j,k\ge 1$ and $i+j+k=6$, is pancyclic. This, together with results by Ferrara, Morris, Wenger, and Ferrara et al. completes a characterization of the graphs Y such that every $\{K_{1,3},Y\}$-free graph is pancyclic. In addition, this represents the best known progress towards answering a question of Gould concerning a characterization of the pairs of forbidden subgraphs that imply pancyclicity in 4-connected graphs.

Introduction

All graphs in this paper are simple, and for a graph G, we often use $|G|$ to denote $|V(G)|$. G is Hamiltonian if it contains a spanning cycle and pancyclic if it contains cycles of each length from 3 to $|G|$. For a positive integer k, G is k-connected if $G-S$ is connected, for all $S\subseteq V(G)$ with $|S|<k$. Given a family $\mathcal{F}$ of graphs, G is said to be $\mathcal{F}$-free if G contains no member of $\mathcal{F}$ as an induced subgraph. If $K_{1,3}\in \mathcal{F}$, then G is said to be claw-free. $L(G)$ is used to denote the line graph of G, in which the vertices of $L(G)$ are the edges of G, and two vertices in $L(G)$ are adjacent exactly when the corresponding edges are incident in G. Note that $L(G)$ is claw-free for every graph G.

The following well-known conjecture of Matthews and Sumner [9] has provided the impetus for a great deal of research into the Hamiltonicity of claw-free graphs.

Conjecture 1 The Matthews-Sumner Conjecture

If G is a 4-connected, claw-free graph, then G is Hamiltonian.

In [10], Ryjáček demonstrated that the Matthews-Sumner conjecture is equivalent to a conjecture of Thomassen [14] that every 4-connected, line graph is Hamiltonian. Also in [10], Ryjáček showed that every 7-connected, claw-free graph is Hamiltonian. Kaiser and Vrána [8] then showed that every 5-connected, claw-free graph with minimum degree at least 6 is Hamiltonian, which currently represents the best general progress towards affirming Conjecture 1. Recently in [13], Conjecture 1 was also shown to be equivalent to the statement that every 4-connected, claw-free graph is Hamiltonian-connected.

The Matthews-Sumner Conjecture has fostered a large body of research into other cycle-structure properties of claw-free graphs. In this paper, we are specifically interested in the pancyclicity of highly connected, claw-free graphs. Significantly fewer results of this type can be found in the literature, in part because it has been shown in many cases [11], [12] that closure techniques like those in [10], shown to be useful to a number of cycle problems, do not apply to pancyclicity. For instance, as shown in [1], for every $k\ge 1$ there exists a k-connected, claw-free graph that is not pancyclic, but has complete Ryjáček closure. This further precludes a Matthews-Sumner type conjecture for pancyclicity.

Thus, we turn our attention to families of forbidden subgraphs that imply pancyclicity in graphs with sufficiently high connectivity. Building upon several prior results, Faudree and Gould gave the following characterization of forbidden pairs that imply pancyclicity in 2-connected graphs. Here $N(i,j,k)$ is the generalized net, which is obtained by identifying one vertex of a triangle with an endpoint of a path of length i, identifying another vertex of the triangle with an endpoint of a path of length j, and the third vertex with an endpoint of a path of length k.

Theorem 1 Faudree, Gould [3]

Let X and Y be connected graphs on at least three vertices. If neither X nor Y is $P_3$ and Y is not $K_{1,3}$, then every 2-connected, $\{X,Y\}$-free graph G is pancyclic if and only if $X=K_{1,3}$ and Y is an induced subgraph of either $P_6$ or $N(2,0,0)$.

Gould, Łuczak and Pfender [6] continued this line of inquiry, and obtained the following characterization of forbidden pairs of subgraphs that imply pancyclicity in 3-connected graphs. Here Ł denotes the graph obtained by connecting two disjoint triangles with a single edge.

Theorem 2 Gould, Łuczak, Pfender [6]

Let X and Y be connected graphs on at least three vertices. If neither X nor Y is $P_3$ and Y is not $K_{1,3}$, then every 3-connected, $\{X,Y\}$-free graph G is pancyclic if and only if $X=K_{1,3}$ and Y is an induced subgraph of one of the graphs in the family

$$\mathcal{F}=\{P_7,\mathrm{Ł},N(i,j,k):i+j+k=4\}.$$

Motivated by the Matthews-Sumner Conjecture and Theorem 1, Theorem 2, Gould posed the following problem at the 2010 SIAM Discrete Math meeting in Austin, TX.

Problem 1

Characterize the pairs of forbidden subgraphs that imply a 4-connected graph is pancyclic.

The first progress towards this problem appears in [5].

Theorem 3 Ferrara, Morris, Wenger [5]

If G is a 4-connected, $\{K_{1,3},P_{10}\}$-free graph, then either G is pancyclic or G is the line graph of the Petersen graph. Consequently, every 4-connected, $\{K_{1,3},P_9\}$-free graph is pancyclic.

The line graph of the Petersen graph is 4-connected, claw-free and contains no cycle of length 4 (see $L(P)$ in Fig. 1). Noting that in Theorem 2, all generalized nets of the form $N(i,j,0)$ with $i+j=4$ are in the family $\mathcal{F}$, Ferrara, Gould, Gehrke, Magnant, and Powell [4] showed the following.

Theorem 4 Ferrara, Gould, Gehrke, Magnant, Powell [4]

Every 4-connected, $\{K_{1,3},N(i,j,0)\}$-free graph with $i+j=6$ is pancyclic. This result is best possible, in that the line graph of the Petersen graph is $N(i,j,0)$-free for all $i,j\ge 0$ with $i+j=7$.

In this paper, we continue the progress towards Problem 1. In particular, we prove the following three theorems, which yield a characterization of those graphs Y such that every 4-connected, $\{K_{1,3},Y\}$-free graph is pancyclic.

Theorem 5

Every 4-connected, $\{K_{1,3},N(i,j,k)\}$-free graph with $i,j,k\ge 1$ and $i+j+k=6$ is pancyclic. This result is best possible, in that the line graph of the Petersen graph is $N(i,j,k)$-free for all $i,j,k\ge 1$ with $i+j+k=7$.

Theorem 6

Let X and Y be connected graphs with at least three edges such that every 4-connected, $\{X,Y\}$-free graph is pancyclic. Then, without loss of generality, X is either $K_{1,3}$ or $K_{1,4}$ and Y is an induced subgraph of one of $P_9$, Ł, or the generalized net $N(i,j,k)$ with $i+j+k=6$.

Theorem 7

Let Y be a connected graph with at least three edges. Every 4-connected, $\{K_{1,3},Y\}$-free graph is pancyclic if and only if Y is an induced subgraph of one of the graphs in the family

$$\mathcal{F}=\{P_9,\mathrm{Ł},N(i,j,k):i+j+k=6\}.$$

Observe that Theorem 7 follows from Theorem 2, Theorem 3, Theorem 4, Theorem 5, Theorem 6. It is also worth noting that results in [7] can be used to prove Theorem 7 for graphs sufficiently large. Thus, part of the goal of this paper is to complete this characterization for all graphs.

The outline of this paper is as follows. We prove Theorem 6 in Section 2, and prove Theorem 5 in Sections 3 - 7. In particular, we show the existence of cycles of lengths 3, 4, and 5 in Section 3. Then in Section 4, we prove several technical lemmas that will be used in Sections 5-7. In Sections 5, 6, and 7, we consider each of the nets $N(2,2,2),N(3,2,1)$, and $N(4,1,1)$, respectively, and show that a 4-connected, claw-free graph that avoids that specific net is pancyclic. Lastly, we present some ideas for future research in Section 8.