Cycle Extendability of Hamiltonian Strongly Chordal Graphs
CCycle Extendability of Hamiltonian Strongly Chordal Graphs
Guozhen Rong ∗ Wenjun Li † Jianxin Wang ∗ Yongjie Yang ‡ July 31, 2020
Abstract
In 1990, Hendry conjectured that all Hamiltonian chordal graphs are cycle extendable. Aftera series of papers confirming the conjecture for a number of graph classes, the conjecture is yetrefuted by Lafond and Seamone in 2015. Given that their counterexamples are not strongly chordalgraphs and they are all only 2-connected, Lafond and Seamone asked the following two questions:(1) Are Hamiltonian strongly chordal graphs cycle extendable? (2) Is there an integer k such that all k -connected Hamiltonian chordal graphs are cycle extendable? Later, a conjecture stronger thanHendry’s is proposed. In this paper, we resolve all these questions in the negative. On the positiveside, we add to the list of cycle extendable graphs two more graph classes, namely, Hamiltonian4-leaf powers and Hamiltonian { FAN , A } -free chordal graphs. A graph is
Hamiltonian if it has a
Hamiltonian cycle . Investigating sufficient conditions for the existenceof a Hamiltonian cycle has been a prevalent topic, initiated by the seminal work of Dirac [8]. Acommonly used scheme to show the existence of a Hamiltonian cycle is to arise a contradiction to theassumption that the graph has no Hamiltonian cycle, by means of extending an assumed maximumnonHamiltonian cycle to a longer cycle. After observing this, Hendry [12, 13, 14] proposed the conceptof cycle extendability. Concretely, a cycle C is extendable if there exists a cycle C (cid:48) which contains allvertices of C plus one more vertex not in C . A graph is cycle extendable if all nonHamiltonian cyclesof the graph are extendable. The notion of cycle extendability is related to the well-studied notion ofpancyclicity. Recall that a graph on n vertices is pancyclic if it contains a cycle of length (cid:96) for every (cid:96) such that 3 (cid:54) (cid:96) (cid:54) n . Clearly, if a graph is cycle extendable then it is pancyclic, and if it is pancyclic thenit is Hamiltonian. In [13], Hendry studied several sufficient conditions for cycle extendable graphs, andin the conclusion he put forward a “reverse” notion of cycle extendability, namely the cycle-reducibility.Precisely, a graph is cycle-reducible if for every cycle C in the graph there exists a cycle C (cid:48) which consistsof | C | − C . In light of the facts that (1) a graph is cycle-reducible if and only if it is achordal graph, and (2) Hamiltonian chordal graphs are pancyclic, Hendry [13] proposed the followingconjecture. Hendry’s Conjecture.
Hamiltonian chordal graphs are cycle extendable. ∗ School of Computer Science and Engineering, Central South University, Changsha, China. [email protected],[email protected] † Hunan Provincial Key Laboratory of Intelligent Processing of Big Data on Transportation, Changsha University of Scienceand Technology, Changsha, China. [email protected] ‡ Faculty of Human and Business Sciences, Saarland University, Saarbr¨ucken, Germany. [email protected] a r X i v : . [ c s . D M ] J u l ecall that chordal graphs are the graphs without holes (induced cycles of length at least four) asinduced subgraphs. Since the work of Hendry [13], the above conjecture has received a considerableamount of attention. Remarkably, Hendry’s conjecture has been confirmed for a number of special graphclasses including such as planar Hamiltonian chordal graphs [15], Hamiltonian interval graphs [2, 7],Hamiltonian split graphs [2], etc. In 2013, Abueida, Busch, and Srithara [1] added to the list of cycleextendable graph classes the Hamiltonian spider intersection graphs, a super class of Hamiltonianinterval and Hamiltonian split graphs.Though all these confirmative works continuously fill the gap step by step and provide moreand more evidence towards a positive answer to Hendry’s conjecture, the conjecture is, somewhatsurprisingly, eventually disproved the first time by Lafond and Seamone [16] in 2015. Particularly,Lafond and Seamone derived a counterexample with only 15 vertices. Based on this counterexample,they also showed that for any n (cid:62)
15, a counterexample on n vertices exists. Neverthelss, Lanfond andSeamone’s work is not the end of the story, as there are many intrusting subclasses of chordal graphsfor which whether Hendry’s conjecture holds stilled remained open. Notably, as all counterexamplesconstructed in [16] contain an induced 3 -sun which is a forbidden induced subgraph of strongly chordalgraphs (see Figure 2 for a 3-sun), and contain at least one degree-2 vertex, Lanfond and Seamoneproposed two questions which we restated as two conjectures as follows. (For notions in the followingdiscussions, we refer to the next section for the formal definitions.) Lafond-Seamone Conjecture 1 (LSC1).
Hamiltonian strongly chordal graph are cycleextendable.
Lafond-Seamone Conjecture 2 (LSC2).
There exists an integer k > such that all k -connected Hamiltonian chordal graphs are cycle extendable. Later, based on the concept of S -cycle extendability first studied in [5], a more general conjecturestronger was proposed. Let S be a nonempty subset of positive integers. A cycle C in a graph G is S -extendable if there exists another cycle C (cid:48) in G which consists of all vertices of C and i additionalvertices for some integer i ∈ S . A graph is S -cycle extendable if every nonHamiltonian cycle of thegraph is S -extendable. Clearly, { } -cycle extendable graphs are exactly cycle extendable graphs. Afterobserving that the counterexamples of Lanfond and Seamone are {
1, 2 } -cycle extendable, Arangno [4]put forward in his Ph.D. thesis the following conjecture. Arangno’s Conjecture.
Hamiltonian chordal graphs are {
1, 2 } -cycle extendable.The first contribution of this paper is the refutation of all of the above conjectures . In fact, werefute more general conjectures. Theorem 1.1.
Let S be a nonempty set of positive integers. Let t be the maximum integer in S . Then, forevery integer n (cid:62) + t , there exists a Hamiltonian strongly chordal graph which is not S -cycle extendable. Our counterexamples are modifications of those derived by Lanfond and Seamones. Regarding k -connected chordal graphs, we have the following result. Theorem 1.2.
For every fixed nonempty subset S of positive integers and every fixed integer k > , thereexists a k -connected Hamiltonian strongly chordal graph that is not S -cycle extendable. On the positive side, we add to the list of graph classes fulfilling Hendry’s conjecture two subclassesof Hamiltonian chordal graphs, namely, Hamiltonian 4-leaf power graphs and Hamiltonian { FAN , A } -free chordal graphs. See Figure 1 for 4- FAN and A . Independent of our work Lafond et al. [20] obtained similar results for these conjectures. a) 4- FAN (b) A (c) K − e Figure 1: 4-
FAN , A , and K − e It is known that k -leaf powers for all integers k are natural subclass of strongly chordal graphs [17].Very recently, Gerek proved that Hendry’s conjecture holds for Ptolemaic graphs. This implies thatHendry’s conjecture holds for k -leaf powers for k =
1, 2, 3 because they are subclasses of Ptolemaicgraphs. We complement this result by following theorem.
Theorem 1.3.
Hamiltonian -leaf powers are cycle extendable. Finally, we confirm Hendry’s conjecture for Hamiltonian { FAN , A } -free chordal graphs. Theorem 1.4.
Hamiltonian { FAN , A } -free chordal graphs are cycle extendable. Note that { FAN , A } -free chordal graphs is a super class of the Ptolemaic graphs. Therefore, ourresult extends the result by Gerek [11]. Additionally, { FAN , A } -free chordal graphs contain also thewell-partitioned chordal graphs coined by Ahn et.al. [3] very recently. Organization.
In Section 2, we provide basic notions used in our paper. Section 3 is devoted to theproofs of Theorems 1.1–1.2, and Section 4 composites the proofs of Theorem 1.3–1.4. We conclude ourstudy in Section 5.
We assume the reader is familiar with the basics of graph theory. We reiterate numerous importantnotions used in our discussions, and refer to [19] for notions use in the paper but whose definitions arenot provided in this section. By convention, for an integer i , we use [ i ] to denote the set of all positiveintegers at most i .All graphs considered in this paper are finite, undirected, and simple. The vertex set and the edgeset of a graph G are denoted by V ( G ) and E ( G ) , respectively. We use uv to denote an edge betweentwo vertices u and v . For a vertex v ∈ V ( G ) , N ( v ) = { u | uv ∈ E ( G ) } denotes the (open) neighborhood of v , and N [ v ] = N ( v ) ∪ { v } denotes the closed neighborhood of v . The degree of v , denoted by d ( v ) , isthe cardinality of N ( v ) . For a subset X ⊆ V ( G ) , let N ( X ) = (cid:83) v ∈ X N ( v ) \ X and N [ X ] = N ( X ) ∪ X . Thesubgraph induced by X is denoted by G [ X ] . For brevity, we write G − X for the subgraph of G inducedby V ( G ) \ X . When X = { x } , we simply use the shorthand G − x for G − X .We say that two vertices are true twins if they have the same closed neighborhood. A vertex v is atrue-twin vertex (true-twin for short) if there is another vertex u such that v and u are true twins. Inthis case, we call v (resp. u ) a true-twin for u (resp. v ). A true-twins pair refers to a pair { u , v } suchthat u and v are true twins. A true-twin class of a graph is a maximal set of vertices that are pairwisetrue twins. It is easy to see that every graph admits a unique partition of its vertex set into true-twinclasses.A vertex is universal in a graph if it is adjacent to all other vertices. A vertex of degree 0 is called an isolated vertex. A vertex v in a graph G is simplicial if N [ v ] is a clique. Moreover, if the vertices in N ( v ) x y x y y (a) 3-sun y y y y x x x x (b) 4-sun Figure 2: 3-sun and 4-suncan be ordered as ( v , v , . . . , v k ) such that N [ v ] ⊆ N [ v ] ⊆ · · · ⊆ N [ v k ] , where k = | N ( v ) | , then wesay that v is simple . A simple vertex is always simplicial.A hole in a graph G is an induced cycle of G of length at least 4. An independent set of G is a subset S of vertices such that G [ S ] contains only isolated vertices. A clique is a subset S of vertices such that thereis an edge between every pair of vertices in S . For an integer k (cid:62)
3, a k -sun is a graph of 2 k verticeswhich can be partitioned into an independent set X = { x , . . . , x k } and a clique Y = { y , . . . , y k } suchthat for every i ∈ [ k ] , x i is only adjacent to y i ( mod k ) and y i + ( mod k ) . See Figure 2 for the 3-sun and4-sun.A graph is chordal if it does not contain any holes. Strongly chordal graphs are chordal graphswithout induced k -suns for all k (cid:62)
3. Hence, the minimal forbidden induced subgraphs of stronglychordal graphs are k -suns and holes, none of which contains a universal vertex or a true-twins pair.Strongly chordal graphs admit an ordering characterization. In particular, a simple elimination ordering of a graph G is an ordering ( v , v , . . . , v n ) over V ( G ) such that for every i ∈ [ n ] , v i is simple in thesubgraph of G induced by { v , . . . , v i } . It has been proved that a graph is a strongly chordal graph ifand only if it admits a simple elimination ordering [9].A graph G is a k -leaf power if there exists a tree T such that (1) the vertices of G one-to-onecorrespond to the leaves of T , and (2) for every two vertices u , v ∈ V ( G ) , it holds that uv ∈ E ( G ) if andonly if the distance between u and v in T is at most k . It is a folklore that k -leaf powers are stronglychordal graphs [17].A graph is connected if it has only one vertex or between every two vertices there exists a path in thegraph. A graph is k -connected if it is connected after the deletion of any subset of at most k − In this section, we explore counterexamples to the conjectures given in Section 1. To give the readera better understanding of our results, we first study counterexamples to LSC1, i.e., we provide theproof for Theorem 1.1 restricted to S = { } . After laying out these results, we show the proofs forTheorems 1.1–1.2. Our counterexample to LSC1 is obtained from the 15-vertex counterexample of Lafond and Seamoneby an addition of one more edge. However, the selection of which edge should be added and thearguments for that the resulting graph is a counterexample to LSC1 are not straightforward.The following two lemmas follows immediately from the definitions of strongly chordal graphs andHamiltonian graphs respectively. 4 bservation 3.1.
A strongly chordal graph remains strongly chordal after adding or deleting a universalvertex or a true-twin for any vertex.
Observation 3.2.
A Hamiltonian graph remains Hamiltonian after adding a universal vertex or true-twinvertex for any vertex.
Our counterexample is based on the graphs (cid:98) H and (cid:98) H − shown in Figure 3, where the graph (cid:98) H isobtained from the 15-vertex counterexample of Lafond and Seamones by adding one edge [16]. abcd e f g u u u v v v v v (a) (cid:98) H abcd e f g u u u v v v v v (b) (cid:98) H − Figure 3: Two graphs (cid:98) H and (cid:98) H − . The graph (cid:98) H − { v , v , v , v , v } is composed by a 7-vertex path abcdefg plus three universal vertices u , u , and u . Heavy edges are in bold. Lemma 3.3. (cid:98) H is a Hamiltonian strongly chordal graph.Proof. The graph (cid:98) H is Hamiltonian since av u v gfv u v bcdev u a is a Hamiltonian cycle of (cid:98) H .It remains to prove that (cid:98) H is a strongly chordal graph. It is fairly easy to check that the vertices v , v , v , v , and v are the simple vertices in (cid:98) H . Let H (cid:48) = (cid:98) H − { v , v , v , v , v } . Since simplevertices are not in any induced holes and k -suns for all k (cid:62)
3, it suffices to prove that H (cid:48) is a stronglychordal graph. To this end, observe that H (cid:48) consists of a 7-vertex path abcdefg and three universalvertices u , u , and u . A path is clearly a strongly chordal graph. Then, by Observation 3.1, H (cid:48) is astrongly chordal graph too.The vertices v , v , v , v , and v are all of degree-2 in (cid:98) H . We call the edges incident to them heavyedges (see Figure 3). We have the following two observations. Observation 3.4.
Every cycle in (cid:98) H containing { v , v , v , v , v } must contain all the heavy edges. Observation 3.5. C is a cycle containing all heavy edges in (cid:98) H if and only if C is a cycle containing allheavy edges in (cid:98) H − . Now we study a nonextendable cycle in (cid:98) H . Lemma 3.6.
The cycle C = av u v gu v efv u v ba in (cid:98) H is not extendable. roof. Note that the cycle C contains all vertices of (cid:98) H except the two vertices c and d . Suppose forcontradiction that C admits an extension C (cid:48) . Clearly, C (cid:48) contains { v , v , v , v , v } , and it holds that C (cid:48) is a Hamiltonian cycle of either (cid:98) H − c or (cid:98) H − d . By Observations 3.4 and 3.5, C (cid:48) is either a Hamiltoniancycle of (cid:98) H − − c or a Hamiltonian cycle of (cid:98) H − − d and, moreover, C (cid:48) contains all heavy edges.If C (cid:48) is a Hamiltonian cycle of (cid:98) H − − c , then as the vertex d has degree 2 in (cid:98) H − − c , the twoedges du and de incident to d are contained in the cycle C (cid:48) . However, these two edges togetherwith the heavy edges incident to v form a cycle du v ed of length four, contradicting that C (cid:48) is aHamiltonian cycle of (cid:98) H − − c .Finally, if C (cid:48) is a Hamiltonian cycle of (cid:98) H − − d , then as c and e are both degree-2 vertices in (cid:98) H − − d , C (cid:48) must contain the four edges bc , cu , ef , and ev . However, these four edges together with the heavyedges bv , v u , u v , fv , and v u form a cycle bv u v fev u cb of length 9, contradiction that C (cid:48) is a Hamiltonian cycle of (cid:98) H − − d .Armed with the above lemmas, we are ready to showcase the proof of Theorem 1.1 restricted to S = { } . Proof of Theorem 1.1 for S = { } . Let n be an integer such that n (cid:62)
15. The counterexample for n =
15 has been discussed. So, we assume that n (cid:62)
16. Based on the graph (cid:98) H , we construct acounterexample to LSC1 of n vertices as follows. Let C be the cycle of (cid:98) H as specified in Lemma 3.6,i.e., C = av u v gu v efv u v ba . Let i and j be two nonnegative integers such that i + j = n − (cid:98) H + be the graph obtained from (cid:98) H by adding i true-twins of v and adding j true-twins of d .Let X and Y denote, respectively, the set of the i true-twins of v and the set of the j true-twinsof d . By Observations 3.1 and 3.2, and Lemma 3.3, (cid:98) H + is a Hamiltonian strongly chordal graph. Let x x · · · x i be any arbitrary but fixed order of X . Obviously, C + = av x . . . x i u v gu v efv u v ba isa nonHamiltonian cycle in (cid:98) H + . In addition, (cid:98) H + − { a , u } is disconnected with { v } ∪ X being one ofits connected components. Hence, in every cycle that contains all vertices of C + , vertices in { v } ∪ X appear consecutively.We claim that the cycle C + is not extendable. Assume for the contradiction that there exists acycle C ∗ in (cid:98) H + such that V ( C ∗ ) = V ( C + ) ∪ { z } , where z ∈ { c , d } ∪ Y . As just discussed, vertices in { v } ∪ X appear consecutively in C ∗ . It follows that the cycle C ∗ without the vertices in X is also a cyclein (cid:98) H + , and hence a cycle in (cid:98) H too. Let C (cid:48) denote this cycle. Due to Lemma 3.6, it holds that z / ∈ { c , d } .Therefore, it must be that z ∈ Y , i.e., z is a true twin of d . Then, by replacing z by d in C (cid:48) we obtain acycle of (cid:98) H which is an extension of the cycle av u v gu v efv u v ba , a contradiction to Lemma 3.6.The proof that (cid:98) H + is not cycle extendable is completed. S -Cycle Nonextendability In this section, we show how to modify the counterexamples in the previous section to prove Theo-rems 1.1 and 1.2.
Proofs of Theorem 1.1.
Let S be a fixed nonempty set of positive integers and let t be the maximuminteger in S , i.e., i (cid:54) t for all i ∈ S . To show that a graph is not S -cycle extendable, we show that thegraph is not [ t ] -cycle extendable. We modify the graph (cid:98) H into a graph G t as follows: We first replacethe edge cd by a ( t + ) -vertex path P = cz z . . . z t − d , and then we add edges so that each z i , where i ∈ [ t − ] , is adjacent to u , u , and u . For convenience, we use z to denote c , use z t to denote d ,and define Z = { z , . . . , z t } . Let G − t be the graph obtained from G t by deleting all nonheavy edgesincident to u or u , and deleting the edge eu . See Figure 4 for illustrations of G t and G − t .6 bz z z z e f g u u u v v v v v (a) G t abz z z z e f g u u u v v v v v (b) G − t Figure 4: Two graphs G t and G − t in the proof of Theorem 1.1 for t = G t is Hamiltonian since av u v gfv u v bz . . . z t ev u a is a Hamiltonian cycle of G t .Analogous to the proof of Lemma 3.3, it can be shown that G t is a Hamiltonian strongly chordal graph.Additionally, Observations 3.4 and 3.5 apply to G t too. That is, the following conditions are fulfilledby G t . • Every cycle in G t containing { v , v , v , v , v } contains all heavy edges. • C is a cycle containing all heavy edges in G t if and only if C is a cycle containing all heavy edgesin G − t .Due to these two observations, to complete the proof, we need only to show that the cycle C = av u v gu v efv u v ba in G − t is not { i } -extendable for all i ∈ [ t ] . We prove this by induction.For the base case where i =
1, our proof goes as follows. Assume for the sake of contradiction that C can be extended to C (cid:48) such that V ( C (cid:48) ) = V ( C ) ∪ { z } where z ∈ Z . Since z j where j ∈ [ t − ] has degreeone in G − t − ( Z \ { z j } ) , it holds that z / ∈ Z \ { z , z t } . Note that G − t − ( Z \ { z } ) is isomorphic to (cid:98) H − − c ,and G − t − ( Z \ { z t } ) is isomorphic to (cid:98) H − − d . Then, by Lemma 3.6, z cannot be either of z and z t too.This completes the proof for the base case, and if t = t > C is not [ i − ] -extendable for any integer i such that 2 (cid:54) i (cid:54) t , weclaim that the cycle C in G − t is not { i } -extendable. We prove this by contradiction. Assume for the sakeof contradiction that C ∗ is an { i } -extension of C in G − t such that V ( C ∗ ) = V ( C ) ∪ Z (cid:48) for some Z (cid:48) ⊆ Z with | Z (cid:48) | = i . Let Z − = Z \ Z (cid:48) . Obviously, Z (cid:48) (cid:54) = { z , z t } since G − t − ( Z \ { z , z t } ) is not Hamiltonian. Notethat vertices of Z are all degree-3 vertices in G − t , Z \ Z (cid:48) (cid:54) = ∅ , and every vertex of Z (cid:48) in C ∗ has degree 2.Hence, there exists at least one vertex of Z (cid:48) in G − t − Z − with degree 2, and there exists no vertex of Z (cid:48) in G − t − Z − with degrees 0 or 1. As Z (cid:48) (cid:54) = { z , z t } , there are two vertices z , z (cid:48) ∈ Z (cid:48) such that z has degree 2in G − t − Z − , and zz (cid:48) is an edge in G − t − Z − . It is known that zu , z (cid:48) u ∈ E ( G − t − Z − ) . Replacing theedges zz (cid:48) and zu by the edge z (cid:48) u in C ∗ , we get a cycle on V ( C ∗ ) \ { z } , an ( i − ) -extension of C , acontradiction.A counterexample of n > + t vertices can be obtained from the above graph G t by adding n − − t true-twins of v .Now, we move on to the proof of Theorem 1.2. We need a few additional notions for our exposition.For two positive integers a and b , an ( a , b ) -star is a graph whose vertex set can be partitioned into an7 bz z z z e f g u u u v v v v v x x x y y y Figure 5: The graph G t . The set of vertices in the left box induces a G t , and { x , x , x , y , y , y } induces a (
3, 3 ) -star. A multi-edge between a vertex and the left box means that the vertex is adjacentto all vertices in the box.independent set X of a vertices and a clique Y of b vertices such that all vertices in X are adjacent to allvertices in Y . Moreover, such a partition ( X , Y ) is called the ( a , b ) -partition of the graph. For a fixedinteger k >
1, we define a graph G kt which is constructed in the following way (see Figure 5 for anillustration of G t ):(1) take the union of G t and S k , where S k is a ( k , k ) -star with the ( k , k ) -partition ( X , Y ) ;(2) for each vertex y ∈ Y , add an edge between y and each vertex in G t ;(3) add an edge between u and each vertex of X . Proof of Theorem 1.2.
Similar to the proof of Theorem 1.1, let S be a fixed nonempty set of positiveintegers and let t be the maximum integer in S . In addition, let G t and Z = { z , . . . , z t } be as defined inthe proof of Theorem 1.1. To prove Theorem 1.2, we first show that the graph G kt is a k -connectedHamiltonian strongly chordal graph, and then we show that the graph G kt contains a cycle which isnot [ t ] -extendable. Obviously, after deleting any arbitrary subset of at most k − Y remains in the graph. Then, as all vertices in Y are universal vertices, it followsthat G kt is k -connected . Let x x . . . x k and y y . . . y k be any arbitrary but fixed orders over X and Y respectively. The following Hamiltonian cycle is the evidence that G t is Hamiltonian. v gfv u v bz z . . . z t ev u av u x y x y . . . x k y k v .Note that after deleting all the universal vertices in Y , vertices of X are all simple vertices. As simplevertices and universal vertices of a graph never participate in any holes or induced j -suns for all integers j (cid:62)
3, it holds that G kt is a strongly chordal graph if G kt − ( X ∪ Y ) = G t is, which is the case as shownin the proof of Theorem 1.1.Now we prove that there is a cycle C in G kt which is not [ t ] -extendable by contradiction. Inparticular, we define C = x y x y . . . x k y k v gu v efv u v bav u x . It is clear that C contains all In fact, one can check that the graph is even k + G kt except z , z , . . . , z t . Suppose for contradiction that C is [ t ] -extendable in G kt . Notethat | Z | = t +
1. Then there exists a nonHamiltonian cycle C (cid:48) of G kt such that V ( C ) ⊂ V ( C (cid:48) ) . Let S = { u } ∪ Y . Let G (cid:48) be the subgraph of G kt induced by V ( C (cid:48) ) , and F = G (cid:48) − X ∪ Y . Clearly, S ⊂ V ( G (cid:48) ) .The graph G (cid:48) − S has exactly k + F − { u } and the k isolated vertices in X .Then, as | S | = k +
1, the removal of S from C (cid:48) yields exactly k + G (cid:48) − S . This implies that the path corresponding to F − { u } is a Hamiltonianpath of F − { u } . Moreover, because v and v are degree-1 vertices in F − u , we know that the twoends of the Hamiltonian path of F − u are v and v . We can then obtain a Hamiltonian cycle C ∗ of F by inserting u into the Hamiltonian path of F − u . Let (cid:98) C = av u v gu v efv u v ba . We haveproven in Theorem 1.1 that (cid:98) C is not [ t ] -extendable in G t . Whereas, F is a proper induced subgraph of G t and V ( (cid:98) C ) ⊂ V ( C ∗ ) , which means that C ∗ is a [ t ] -extension of (cid:98) C in G t , a contradiction. This section is devoted to the proofs of Theorem 1.3 and 1.4. A first step to study Hendry’s conjecturefor a class of graphs has commonly been to gain the characterization of the graphs by forbiddenstructures. Unfortunately, to the best of our knowledge, the full characterization of 4-leaf powers byforbidden induced subgraphs still remained as a challenging open question so far. Nevertheless, 4-leafpowers without true twins have been fully characterized [18]. Our study needs only two forbiddeninduced subgraphs of 4-leaf powers without true twins, as stated in the following lemma.
Lemma 4.1. [18] Every -leaf power graph without true twins does not contains any FAN or K − e asinduced subgraphs (see FAN and K − e in Figure 1). Based on the above lemma, for 4-leaf powers that may contain true twins, we have the followinglemma.
Lemma 4.2.
Let G be a -leaf power. Then(i) G contains no FAN as an induced subgraph, and(ii) every induced K − e in G must contain a true-twins pair of G .Proof. Clearly, 4-
FAN does not contain any true-twins pairs. Then, by Lemma 4.2, a 4-
FAN is not a 4-leafpower. Since 4-leaf powers are hereditary (closed under vertex deletions), G contains no 4- FAN as aninduced subgraph. So, Statement (i) holds.Statement (ii) is vacuously true if G contains no induced K − e . Let F be an induced K − e in G . Assume for contradiction that F does not contain any true-twins pairs in G . Let G (cid:48) be the graphgenerated from G by deleting some vertices as following: for each true-twin class S of G , if F containsone vertex of S , delete S \ V ( F ) from G ; otherwise, delete | S | − S arbitrarily. As 4-leafpowers are hereditary, the graph G (cid:48) remains a 4-leaf power. Moreover, the graph G (cid:48) does not containany true-twins pairs anymore. However, G (cid:48) contains F , a K − e , as an induced subgraph, contradictingwith Lemma 4.1.The following lemmas are from [16]. Lemma 4.3. [16] Let C be a cycle of a chordal graph and let uv an edge in C . Then u and v have acommon neighbor in V ( C ) . Lemma 4.4. [16] Let G be a connected chordal graph and S a clique of V ( G ) . If G − S is disconnected,then each connected component of G − S contains a simplicial vertex of G . G − S is connected. Corollary 4.5.
Let G be a connected chordal graph and S a clique of V ( G ) . Then each connected componentof G − S contains a simplicial vertex of G .Proof. By Lemma 4.4, the statement holds when G − S is disconnected. It remains to prove for the casewhere G − S is connected. Let G (cid:48) be the graph obtained from G by adding a new vertex v which isexactly adjacent to every vertex in S . Apparently, v is simplicial. As a simplicial vertex is not in anyhole, G (cid:48) is also a chordal graph. Now, G (cid:48) − S is disconnected with G − S being one of its connectedcomponents. Then, in light of Lemma 4.4, G − S contains a simplicial vertex u of G (cid:48) , which is clearly asimplicial vertex of G too. Lemma 4.6.
Let n be an integer such that all Hamiltonian chordal graphs of at most n − verticesare cycle extendable, and there exists a Hamiltonian chordal graph G of n vertices which is not cycleextendable. Let C be a nonextendable cycle of G , Q a connected component of G − C , and S = N ( Q ) . Thenthe following conditions hold.(i) S is not a clique of G ,(ii) any two vertices of S are not adjacent in C , and(iii) if two vertices of S are adjacent in G , then they have a common neighbor in Q .Proof. (i) If S is a clique of G , by Corollary 4.5, Q has a simplicial vertex, say v . Then G − v isHamiltonian since G is Hamiltonian and all neighbors of v are pairwise adjacent in G . As allHamiltonian chordal graphs with vertices less than n are cycle extendable, C is extendable in G − v . This implies that C is also extendable in G , a contradiction.(ii) For the sake of contradiction, assume that there are two vertices x , y ∈ S such that xy ∈ E ( C ) .As S = N ( Q ) , both x and y have neighbors in Q . However, x and y cannot have a commonneighbor v in Q : if this was the case, adding v to C and replacing the edge xy by the two edges xv and vy in C yields an extension of C , contradicting that C is not extendable in G . Now, thereexists two distinct vertices x (cid:48) and y (cid:48) such that (1) x (cid:48) is a neighbor of x in Q ; (2) y (cid:48) is a neighborof y in Q ; and (3) x (cid:48) and y (cid:48) have the shortest distance in G [ Q ] among all two distinct verticesfulfilling the first two conditions. However, every shortest x (cid:48) - y (cid:48) path in G [ Q ] plus the threeedges xx (cid:48) , yy (cid:48) , and xy yield a hole, a contradiction.(iii) By (ii), it holds that xy / ∈ E ( C ) . Assume for contradiction that x and y have no common neighborsin Q . Then, analogous to the above proof for (ii), there are two distinct vertices x (cid:48) and y (cid:48) satisfying the same three conditions given above. However, any shortest x (cid:48) - y (cid:48) path in G [ Q ] plusthe three edges xx (cid:48) , yy (cid:48) , and xy yield a hole, a contradiction.We stress that the integer n stipulated in the above lemma must exist, because any Hamiltonianchordal graph of at most four vertices are cycle extendable and there are nonextendable Hamiltonianchordal graphs of 15 vertices [16]. Now, we are ready to prove Theorem 1.3 and 1.4. Proof of Theorem 1.3.
The statement is true for Hamiltonian 4-leaf powers of at most 4 vertices. Sup-pose for contradiction that there exists a Hamiltonian 4-leaf power which is not cycle extendable. Let G be such a graph with the minimum number of vertices. Let C be a nonHamiltonian cycle in G whichis not extendable, and let Q be a connected component of G − C , and S = N ( Q ) . Clearly, S ⊆ V ( C ) .10y Lemma 4.6 (i), S is not a clique. Let x , y ∈ S be two nonadjacent vertices in S with the shortestdistance on C among all pairs of nonadjacent vertices in S . Let P be a shortest x - y path on C . Theremust be at least one inner vertex of P that is contained in S , since otherwise a shortest x - y path in G [ Q ∪ { x , y } ] plus P yield a hole. We break the discussion into two cases. Case 1: there is exactly one inner vertex of P contained in S , say z .By the selection of x and y , we have that xz , yz ∈ E ( G ) . By Lemma 4.6 (ii), xz , yz / ∈ E ( C ) . Let P bethe path between x and z on P , and P the path between y and z on P . By Lemma 4.3, there exists aninner vertex x (cid:48) of P adjacent to x and z , and an inner vertex y (cid:48) of P adjacent to y and z . Since z is theonly inner vertex of P in S , it holds that x (cid:48) , y (cid:48) / ∈ S , i.e., x (cid:48) and y (cid:48) are not adjacent to any vertex of Q .Moreover, it holds that xy (cid:48) , x (cid:48) y , x (cid:48) y (cid:48) / ∈ E ( G ) , since otherwise the shortest x - y path in G [ { x , x (cid:48) , y , y (cid:48) } ] plus a shortest x - y path in G [ Q ∪ { x , y } ] is hole. By Lemma 4.6 (iii), x and z have a common neighbor u in Q , and y and z have a common neighbor v in Q . If x , y , and z have a common neighbor in Q ,(i.e., we can choose u and v such that u = v ,) then { x (cid:48) , x , u , y , y (cid:48) , z } induces a 4- FAN , in which z hasdegree 5. Hence, u and v cannot be the same. Now, we can choose u and v to be the pair such that u and v have the shortest distance in G [ Q ] among all pairs of vertices in Q . Let P (cid:48) be a shortest pathbetween u and v in G [ Q ] . Since uz , vz ∈ E ( G ) , to avoid a hole, all inner vertices of P (cid:48) (if exist) areadjacent to z . Moreover, by the selection of u and v , all inner vertices of P (cid:48) (if exist) are adjacent toneither x nor y . If P (cid:48) has only one edge, then { x (cid:48) , x , u , v , y , z } induces a 4- FAN . If P (cid:48) has more than oneedge, we choose u (cid:48) to be the vertex next to u on P (cid:48) , and v (cid:48) the vertex next to u (cid:48) on P (cid:48) . ( v (cid:48) is v when P (cid:48) is of length 2.) Then, { x (cid:48) , x , u , u (cid:48) , v (cid:48) , z } induces a 4- FAN . Case 2: there are more than one inner vertex of P in S .Let z and z (cid:48) be two inner vertices of P in S such that z and z (cid:48) are the nearest to x and y on P respectively, in terms of the length of the shortest paths between them. Clearly, z and z (cid:48) are distinct. Bythe selection of x and y , we have xz , xz (cid:48) , yz , yz (cid:48) , zz (cid:48) ∈ E ( G ) . By Lemma 4.6 (ii), xz , zz (cid:48) , yz (cid:48) / ∈ E ( C ) . ByLemma 4.3, there exists a vertex x (cid:48) adjacent to x and z , located between x and z on P . Symmetrically,there exists a vertex y (cid:48) adjacent to y and z (cid:48) , located between y and z (cid:48) on P . By the selection of z and z (cid:48) , we have that x (cid:48) and y (cid:48) are not adjacent to any vertex of Q , i.e., x (cid:48) , y (cid:48) / ∈ S . Moreover, it mustbe that xy (cid:48) , x (cid:48) y , x (cid:48) y (cid:48) / ∈ E ( G ) , since otherwise there will be an x - y path whose inner vertices are notadjacent to any vertex of Q and, moreover, this path plus a shortest x - y path of G [ Q ∪ { x , y } ] is a hole.We claim that x , y , and z have no common neighbor in Q . For contradiction, suppose they have acommon neighbor w ∈ Q . Then wz (cid:48) ∈ E ( G ) ; otherwise, { x , w , y , z (cid:48) } induces a hole. Now, { z , z (cid:48) , x , w , y } induces a K − e , with xy being the missing edge. By Lemma 4.2 (ii), there exists a true-twins pair of G from { z , z (cid:48) , x , w , y } . Notice that true-twins pair in G is also a true-twins pair in any induced subgraphcontaining the true-twins pair. This restricts our focus only to the pairs { z , w } , { z (cid:48) , w } , and { z , z (cid:48) } . Thevertices z and w cannot be true twins, since otherwise the vertex next to z in P is also adjacent to w ,contradicting with Lemma 4.6 (ii). Symmetrically, z (cid:48) and w cannot be true twins either. If z and z (cid:48) aretrue twins, then zy (cid:48) ∈ E ( G ) as z (cid:48) y (cid:48) ∈ E ( G ) . We arrive at a contradiction that { x (cid:48) , x , w , y , y (cid:48) , z } inducesa 4- FAN , in which z is the degree 5 vertex. Therefore, z and z (cid:48) are not true twins, and the proof for theclaim that x , y , and z have no common neighbor in Q is completed.Now, we choose u to be a common neighbor of x and z , and v to be the common neighbor of y and z with restriction that u and v have the shortest distance in G [ Q ] . According to Lemma 4.6 (iii) and xz , yz ∈ E ( G ) , the vertices u and v exist. Let P (cid:48) be a shortest u - v path in G [ Q ] . Since uz , vz ∈ E ( G ) , toavoid a hole, all inner vertices of P (cid:48) (if exist) are adjacent to z . By the selection of u and v , all innervertices of P (cid:48) (if exist) are not adjacent to x and y . If P (cid:48) has only one edge, i.e., uv , then { x (cid:48) , x , u , v , y , z } induces a 4- FAN . Hence P (cid:48) has length more than one. We choose u (cid:48) to be the vertex next to u on P (cid:48) ,and v (cid:48) the vertex next to u (cid:48) on P (cid:48) . ( v (cid:48) is v when P (cid:48) is of length 2.) Then, { x (cid:48) , x , u , u (cid:48) , v (cid:48) , z } will inducea 4- FAN , a contradiction. 11 roof of Theorem 1.4.
We use the assumptions analogous to the proof of Theorem 1.3. Similarly,according to the number of inner vertex of P in S , we break it into two cases to discuss. Case 1: there is exactly one inner vertex of P in S , say z . Case 1 is exactly the same as the case 1 inthe proof of Theorem 1.3. Case 2: there are more than one inner vertex of P in S .Let z and z (cid:48) be two inner vertices of P in S such that z and z (cid:48) are the nearest to x and y on P respectively. Clearly, z and z (cid:48) are distinct. By the selection of x and y , we have xz , xz (cid:48) , yz , yz (cid:48) , zz (cid:48) ∈ E ( G ) . By Lemma 4.6(ii), xz , zz (cid:48) , yz (cid:48) / ∈ E ( C ) . By Lemma 4.3, there exists a vertex x (cid:48) adjacent to x and z , located between x and z on P . Symmetrically, there exists a vertex y (cid:48) adjacent to y and z (cid:48) , locatedbetween y and z (cid:48) on P . By the selection of z and z (cid:48) , we have that x (cid:48) and y (cid:48) are not adjacent to anyvertex of Q , i.e., x (cid:48) , y (cid:48) / ∈ S . Moreover, xy (cid:48) , x (cid:48) y , x (cid:48) y (cid:48) / ∈ E ( G ) . Otherwise, there will be a x - y path withinner vertices not adjacent to any vertex of Q , and this path plus a shortest x - y path of G [ Q ∪ { x , y } ] isa hole.We claim that x , y and z have no common neighbor in Q . For contradiction, suppose they have acommon neighbor w ∈ Q . Then wz (cid:48) ∈ E ( G ) ; Otherwise { x , w , y , z (cid:48) } induces a hole. • Case 2.1: if x (cid:48) z (cid:48) ∈ E ( G ) , then { x (cid:48) , x , w , y , y (cid:48) , z (cid:48) } induces a 4- FAN , in which z (cid:48) is the degree 5vertex. • Case 2.2: if y (cid:48) z ∈ E ( G ) , then { x (cid:48) , x , w , y , y (cid:48) , z } induces a 4- FAN , in which z is the degree 5 vertex. • Case 2.3: if x (cid:48) z (cid:48) , y (cid:48) z / ∈ E ( G ) , then { x (cid:48) , x , z , z (cid:48) , y , y (cid:48) } induces a A in which z , z (cid:48) are the degree 4vertices.Therefore, x , y and z have no common neighbor in Q .Now, we choose u to be a common neighbor of x and z , and v to be the common neighbor of y and z with restriction that u and v have the shortest distance in G [ Q ] . By Lemma 4.6(iii) and xz , yz ∈ E ( G ) ,the vertices u and v exist. Let P (cid:48) be a shortest u - v path in G [ Q ] . Since uz , vz ∈ E ( G ) , to avoid a hole,all inner vertices of P (cid:48) (if exist) are adjacent to z . By the selection of u and v , all inner vertices of P (cid:48) (if exist) are not adjacent to either of x and y . If P (cid:48) has only one edge, i.e. uv , then { x (cid:48) , x , u , v , y , z } induces a 4- FAN . Hence P (cid:48) has length more than one. We choose u (cid:48) to be the vertex next to u on P (cid:48) ,and v (cid:48) the vertex next to u (cid:48) on P (cid:48) . ( v (cid:48) is v when P (cid:48) is of length 2.) Then, { x (cid:48) , x , u , u (cid:48) , v (cid:48) , z } will inducea 4- FAN , a contradiction.
In 1990, Hendry proposed the conjecture that every Hamiltonian chordal graph is cycle extendable.Since then, many researchers have devoted themselves into this conjecture, offering a number ofsubclasses of Hamiltonian chordal graphs for which the conjecture holds. However, after 25 years, thisconjecture was eventually refuted by Lanfond and Seamone [16] who crafted a series of counterex-amples, with the smallest one consisting of only 15 vertices. Without ending the story, Lanfond andSeamones proposed two new conjectures (LSC1 and LSC2). Later, Arangno [4] proposed a strongerconjecture. In this paper, we refuted all these conjectures by providing many counterexamples that evensatisfy further conditions (Theorems 1.1–1.2). To complement these negative results, we confirmedHendry’s conjecture for Hamiltonian 4-leaf powers and Hamiltonian { FAN , A } -free chordal graphs.For future research, it is interesting to study whether Hendry’s conjecture holds for Hamiltonian5-leaf powers. A more challenging work would be to identify the maximum odd (resp. even) integer k such that Hendry’s conjecture holds for Hamiltonian k -leaf powers. We reiterate that in general i -leafpowers are ( i + ) -leaf powers, and for each i ∈ [ ] , i -leaf powers are ( i + ) -leaf powers. However,there are 4-leaf powers which are not 5-leaf powers. For more details, we refer to [6, 10].12 eferences [1] A. Abueida, A. Busch, and R. Sritharan. Hamiltonian spider intersection graphs are cycleextendable. SIAM J. Discret. Math. , 27(4):1913–1923, 2013. doi:10.1137/130914164 .[2] A. Abueida and R. Sritharan. Cycle extendability and Hamiltonian cycles in chordal graph classes.
SIAM J. Discret. Math. , 20(3):669–681, 2006. doi:10.1137/S0895480104441267 .[3] J. Ahn, L. Jaffke, O-J. Kwon, and P. de Lima. Well-partitioned chordal graphs: Obstruction setand disjoint paths. In
WG 2020 (to appear) , 2020.[4] D. C. Arangno.
Hamiltonicity, Pancyclicity, and Cycle Extendability in Graphs . PhD thesis, UtahState University, 2014.[5] L. Beasley and D. Brown. On cycle and bi-cycle extendability in chordal and chordal bipartitegraphs.
Congressus Numerantum , 174:41–47, 2005.[6] A. Brandst¨adt and V. B. Le. Structure and linear time recognition of 3-leaf powers.
Inf. Process.Lett. , 98(4):133–138, 2006. doi:10.1016/j.ipl.2006.01.004 .[7] G. Chen, R. J. Faudree, R. J. Gould, and M. S. Jacobson. Cycle extendability of Hamiltonianinterval graphs.
SIAM J. Discret. Math. , 20(3):682–689, 2006. doi:10.1137/S0895480104441450 .[8] G. A. Dirac. Some theorems on abstract graphs.
Proceedings of the London Mathematical Society ,(1):69–81, 1952.[9] M. Farber. Characterizations of strongly chordal graphs.
Discret. Math. , 43(2-3):173–189, 1983. doi:10.1016/0012-365X(83)90154-1 .[10] M. R. Fellows, D. Meister, F. A. Rosamond, R. Sritharan, and J. A. Telle. Leaf powers andtheir properties: Using the trees. In
ISAAC , volume 5369, pages 402–413. Springer, 2008. doi:10.1007/978-3-540-92182-0\_37 .[11] A. Gerek.
Hendry’s Conjecture on Chordal Graph Subclasses . PhD thesis, Lehigh University, 2017.[12] G. R. T. Hendry. Extending cycles in directed graphs.
J. Comb. Theory, Ser. B , 46(2):162–172,1989. doi:10.1016/0095-8956(89)90042-7 .[13] G. R. T. Hendry. Extending cycles in graphs.
Discret. Math. , 85(1):59–72, 1990. doi:10.1016/0012-365X(90)90163-C .[14] G. R. T. Hendry. Extending cycles in bipartite graphs.
J. Comb. Theory, Ser. B , 51(2):292–313,1991. doi:10.1016/0095-8956(91)90044-K .[15] T. Jiang. Planar Hamiltonian chordal graphs are cycle extendable.
Discret. Math. , 257(2-3):441–444, 2002. doi:10.1016/S0012-365X(02)00505-8 .[16] M. Lafond and B. Seamone. Hamiltonian chordal graphs are not cycle extendable.
SIAM J. Discret.Math. , 29(2):877–887, 2015. doi:10.1137/13094551X .[17] R. Nevries and C. Rosenke. Towards a characterization of leaf powers by clique arrangements.
Graphs and Combinatorics , 32(5):2053–2077, 2016. doi:10.1007/s00373-016-1707-x .[18] D. Rautenbach. Some remarks about leaf roots.
Discret. Math. , 306(13):1456–1461, 2006. doi:10.1016/j.disc.2006.03.030 . 1319] D. B. West.
Introduction to Graph Theory . Prentice-Hall, 2000.[20] M. Lafond, B. Seamone, and R. Sherkati. Further results on Hendry’s Conjecture.arXiv:2007.07464 [math.CO], 2020. https://arxiv.org/abs/2007.07464https://arxiv.org/abs/2007.07464