A graph G is called k-ordered if for every sequence of k distinct vertices there is a cycle traversing these vertices in the given order. In the present paper we consider two novel generalizations of this concept, k-vertex-edge-ordered and strongly k-vertex-edge-ordered. We prove the following results for a chordal graph G: (a) G is (2k-3)-connected if and only if it is k-vertex-edge-ordered (k ≥ 3). (b) G is (2k-1)-connected if and only if it is strongly k-vertex-edge-ordered (k ≥ 2). (c) G is k-linked if and only if it is (2k-1)-connected

Böhme, Thomas

Gerlach, Tobias

Stiebitz, Michael

Discussiones Mathematicae Graph Theory

English

Thomas Böhme

Tobias Gerlach

Michael Stiebitz

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Ordered and linked chordal graphs

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   TU Ilmenau | Universitätsbibliothek | ilmedia, 2020 http://www.tu-ilmenau.de/ilmedia Böhme, Thomas; Gerlach, Tobias; Stiebitz, Michael: Ordered and linked chordal graphs URN: urn:nbn:de:gbv:ilm1-2020200158  Original published in: Discussiones mathematicae. Graph theory / Uniwersytet Zielonogórski, Wydział Matematyki, Informatyki i Ekonometrii. - Warsaw : De Gruyter Open. - 28 (2008), 2, p. 367-373. Original published: 2008 ISSN: 2083-5892 DOI: 10.7151/dmgt.1412 [Visited: 2020-01-17]     This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported license.  To view a copy of this license, visit  http://creativecommons.org/licenses/BY-NC-ND/3.0/ Discussiones Mathematicae 367Graph Theory 28 (2008 ) 367–373NoteORDERED AND LINKED CHORDAL GRAPHSThomas Bo¨hme, Tobias GerlachandMichael StiebitzInstitut fu¨r MathematikTechnische Universita¨t IlmenauIlmenau, Germanye-mail: tboehme@theoinf.tu-ilmenau.dee-mail: tobias.gerlach@tu-ilmenau.dee-mail: stieb@mathematik.tu-ilmenau.deAbstractA graph G is called k-ordered if for every sequence of k distinctvertices there is a cycle traversing these vertices in the given order.In the present paper we consider two novel generalizations of thisconcept, k-vertex-edge-ordered and strongly k-vertex-edge-ordered . Weprove the following results for a chordal graph G:(a) G is (2k − 3)-connected if and only if it is k-vertex-edge-ordered(k ≥ 3).(b) G is (2k− 1)-connected if and only if it is strongly k-vertex-edge-ordered (k ≥ 2).(c) G is k-linked if and only if it is (2k − 1)-connected.Keywords: paths and cycles, connectivity, chordal graphs.2000 Mathematics Subject Classification: 05C38, 05C40.1. Introduction and ResultsAll graphs considered in this paper are finite, undirected, and simple, i.e.,without loops or multiple edges. For terminology not defined here werefer to [2]. A graph is chordal if it contains no induced cycles other368 T. Bo¨hme, T. Gerlach and M. Stiebitzthan triangles, and it is called k-linked if for every set of k distinct pairsL = {(s0, t0), . . . , (sk−1, tk−1)} of vertices it contains k internally disjointpaths P0, . . . , Pk−1 such that Pi links si to ti for all i ∈ {0, . . . , k − 1}.We shall call the subgraph of G formed by the union of P0, . . . , Pk−1 an L-linkage. Jung [5] and, independently, Larman and Mani [6] proved that forevery k there is an (minimal) f(k) such that every f(k)-connected graph isk-linked. Bolloba´s and Thomason [1] showed that f(k) ≤ 22k. Recently, itwas proved by Thomas and Wollan [8] that f(k) ≤ 10k. Our second result,Theorem 1.2 below, shows that for the special case of chordal graphs theprecise value of f(k) is 2k − 1.A graph is called k-ordered if for every sequence (v0, . . . , vk−1) of kdistinct vertices there is a cycle of G that contains v0, . . . , vk−1 in the givenorder. This concept was introduced by Ng and Schultz [7], and a survey ofresults on k-ordered graphs is given in [4]. It is easy to see that being k-linked implies being k-ordered. We generalize the concept of k-orderabilityas follows. Let T = (a0, . . . , ak−1) be a sequence of k distinct vertices and/oredges, and let V (T ) and E(T ) denote the sets of vertices and edges in T ,respectively. Let W (T ) denote the set of all vertices that are either containedin T or incident to an edge in T . A T -cycle is a cycle in G that containsa0, . . . , ak−1 in the given order. The sequence T is said admissible if itsatisfies the following conditions.(1) If an edge ai ∈ E(T ) is incident to a vertex aj ∈ V (T ), then |i− j| ≡ 1(mod k).(2) If two edges ai, aj ∈ E(T ) meet in a vertex x /∈ V (T ), then |i − j| ≡ 1(mod k).A graph is called k-vertex-edge-ordered if for every admissible sequence T =(a0, . . . , ak−1) of k distinct vertices and/or edges there is a T -cycle.Theorem 1.1. Let G be a chordal graph on at least 2k − 2 vertices withk ≥ 3. Then the following two statements are equivalent:(a) G is (2k − 3)-connected.(b) G is k-vertex-edge-ordered.Theorem 1.1 implies a conjecture of Faudree [4] for the special case of chordalgraphs.We further generalize this concept. An orientation of an edge e = {u, v}is a pair (u, v); u is called the tail and v the head. Let (a0, . . . , ak−1) be anOrdered and Linked Chordal Graphs 369admissible sequence of k distinct vertices and/or edges. An orientation ofthe edges in this sequence is admissible if it satisfies the following conditions.(3) If the vertex ai is the tail of the edge aj , then i ≡ j − 1 (mod k).(4) If the vertex ai is the head of the edge aj , then i ≡ j + 1 (mod k).(5) If two edges ai, aj ∈ E(T ) meet in a vertex x /∈ V (T ) and j ≡ i +1 (mod k), then x is the head of ai and the tail of aj .A graph is called strongly k-vertex-edge-ordered if for every admissible se-quence T = (a0, . . . , ak−1) of k distinct vertices and/or edges and everyadmissible orientation of the edges of this sequence there is a cycle C of Gthat can be traversed such that a0, . . . , ak−1 are encountered in the givenorder and every edge is traversed according to its orientation, i.e., from tailto head. Clearly, C is a T -cycle.Theorem 1.2. Let G be a chordal graph on at least 2k vertices. Then thefollowing three statements are equivalent:(a) G is (2k − 1)-connected.(b) G is k-linked.(c) G is strongly k-vertex-edge-ordered.2. ProofsLet G be a graph and let x be a vertex of G. Then N(x) denotes the setof all vertices adjacent to x in G. A vertex x of a graph G is simplicialif the subgraph G[N(x)] of G induced by N(x) is complete. The followingProposition 2.1 is a consequence of a well-known theorem of Dirac [3].Proposition 2.1. Let G be a k-connected chordal graph. Then the followinghold:(a) There is a simplicial vertex x ∈ V (G), and G− x is chordal.(b) If G is not complete and x is a simplicial vertex of G, then G − x isk-connected.The following Proposition 2.2 will be frequently used in the proof of Theorem1.1. Its easy proof is left to the reader.Proposition 2.2. Let G be a graph, T = (a0, . . . , ak−1) be an admissible se-quence of distinct vertices and/or edges, X ⊆ V (G), and J ⊆ {0, . . . , k−1}.370 T. Bo¨hme, T. Gerlach and M. StiebitzIf for every vertex x ∈ X there is a j ∈ J such that either x = aj or x isincident to the edge aj , then |X| ≤ 2|J |.Proof of Theorem 1.1. To show that (a) implies (b), we apply inductionon |G|. Let T = (a0, . . . , ak−1) be an admissible sequence. If G is completethe statement of the theorem is clearly true. Hence we may assume that Gis not complete and, therefore, |G| ≥ 2k − 1. By Proposition 2.1, there isa simplicial vertex u ∈ V (G) and G − u is (2k − 3)-connected and chordal.Note that |N(u)| ≥ 2k− 3. Let H = G[N(u)∪{u}]. Clearly, H is complete.Consequently, the assertion is true if W (T ) ⊆ V (H). So, we henceforthassume thatW (T ) 6⊆ V (H).(1)If u /∈ W (T ), then we apply the induction hypothesis to G − u, and weare done. If u ∈ W (T ), then we construct an admissible sequence T ′ =(a′0, . . . , a′k−1) of vertices and/or edges of G − u. Hence, by the inductionhypothesis, there is a T ′-cycle C ′ in G− u. It is easy to see that C ′ can beextended to a T -cycle C in G. For the construction of T ′ we distinguish thefollowing cases.Case 1. u ∈ V (T ), say u = a0.Case 1.1. u is incident to an edge in T , say a1.By Proposition 2.2 and (1), N(u) \ W (T ) 6= ∅. Let v ∈ N(u) be the endof a1 and w ∈ N(u) \ W (T ). Put a′0 = w, a′1 = {v, w}, and a′i = ai fori ∈ {2, . . . , k− 2}. If ak−1 is an edge incident to u, then let x ∈ N(u) be theend of ak−1 and put a′k−1 = {w, x}. Otherwise, let a′k−1 = ak−1.Case 1.2. u is not incident with any edge in T .If |N(u) \ W (T )| ≥ 2, then let v, w ∈ N(u) \ W (T ), and put a′0 = {v, w}and a′i = ai for all i ∈ {1, . . . , k − 1}. If |N(u) \ W (T )| ≤ 1, then there isa vertex v ∈ N(u) such that either v = aj or v is incident to the edge ajand to no other edge in T where j ∈ {1, k − 1}. If not, then all verticesbut at most one in N(u) are either in V (T ) \ {a0, a1, ak−1} or incident to anedge in E(T ) \ {a1, ak−1}. By Proposition 2.2 this implies that |N(u)|− 1 ≤2(k − 3) < 2k − 4, contradicting |N(u)| ≥ 2k − 3. W.l.o.g., we may assumethat j = 1. If |N(u) \ W (T )| = 1, then let w ∈ N(u) \ W (T ) and puta′0 = {v, w} and a′i = ai for all i ∈ {1, . . . , k − 1}. If |N(u) \ W (T )| = 0,Ordered and Linked Chordal Graphs 371then a1 is an edge. If not, then a1 = v and therefore, by Proposition 2.2,W (T ) = N(u) ∪ {u}, contradicting (1). In a similar way it can be shownthat there is a vertex w ∈ N(u) \ {v} such that either w = ak−1 or w isincident to the edge ak−1 and to no other edge in T . Put a′0 = {v, w} anda′i = ai for all i ∈ {1, . . . , k − 1}.Case 2. u /∈ V (T ).Case 2.1. u is incident to two edges in T , say to a0, ak − 1.Let v ∈ N(u) be the end of a0, and w ∈ N(u) be the end of ak−1. If|N(u) \W (T )| ≥ 1, then let x ∈ N(u) \W (T ), and put a′0 = {v, x}, a′k−1 ={x,w}, and a′i = ai for i ∈ {1, . . . , k − 2}. If |N(u) \ W (T )| = 0, then, byProposition 2.2, v 6= a1 and w 6= ak−2. Put a′0 = v, a′k−1 = {v, w} anda′i = ai for i ∈ {1, . . . , k − 2}.Case 2.2. u is incident to exactly one edge in T , say to a0.Let v ∈ N(u) be the end of a0. If |N(u) \ W (T )| ≥ 1, then let w ∈N(u) \ W (T ) and put a′0 = {v, w} and a′i = ai for i ∈ {1, . . . , k − 1}. If|N(u)\W (T )| = 0, then it follows by Proposition 2.2 and (1) that v 6= a1 andv 6= ak−1. By the essentially the same arguments as in Case 1.2 it follows,that if v /∈ V (T ) and v is not incident to any edge in E(T )\{a0}, then thereis a vertex w ∈ N(u) \ {v} such that either w = aj or w is incident to theedge aj and to no other edge in T where j ∈ {1, k − 1}. We may assumew.l.o.g. that j = 1. Put a′0 = {v, w} and a′i = ai for i ∈ {1, . . . , k − 1}. If vis incident to an edge in E(T ) \ {a0}, say a1, then there is a vertex w suchthat either w = ak−1 or w is incident to the edge ak−1 and to no other edgein T . Put a′0 = {v, w} and a′i = ai for i ∈ {1, . . . , k − 1}.Next, we prove that (b) implies (a). It is clear that every k-vertex-edge-ordered graph is connected. Let G be a connected chordal graph on at least2k − 2 vertices that is not (2k − 3)-connected. G has a minimal separatorS ⊆ V (G) with |S| ≤ 2k − 4. Let G1, G2 be two distinct components ofG− S. Since G is chordal, the subgraph H of G induced by S is complete.Let Z = {a1, . . . , ar−2} be a collection of vertices and/or edges in H suchthat Z is a perfect matching of H if |H| is even and a maximal matchingplus the (only) unsaturated vertex, otherwise. Note that r ≤ k. Let T =(a0, . . . , ar−1) where a0 ∈ V (G1) and ar−1 ∈ V (G2). It is not hard to seethat there is no T -cycle in G. Hence every k-vertex-edge-ordered chordalgraph with at least 2k − 2 vertices is (2k − 3)-connected.372 T. Bo¨hme, T. Gerlach and M. StiebitzProof of Theorem 1.2. To show that (a) implies (b), we apply induction on|G|. Since G is (2k−1)-connected, |G| ≥ 2k. If |G| = 2k, then G is complete,and hence it is k-linked. If |G| > 2k, then it follows from Proposition 2.1that G has a simplicial vertex x and G−x is (2k−1)-connected and chordal.Let L = {(s0, t0), . . . , (sk−1, tk−1)} be a set of k distinct pairs of vertices ofG. Let l denote the number of pairs in L containing x. If l = 0, we applythe induction hypothesis to G − x, and we are done. We may thereforeassume that l ≥ 1, say x = s0 = . . . = sl−1. Let A = {t0, . . . , tl−1}, andsuppose that A′ = {t0, . . . , tm−1} = A∩N(x). If there is a ti ∈ A such thatti = x, then suppose that i = l − 1. Consequently, A′′ = A \ (A′ ∪ {x}) ={tm, . . . , tn−1} where n = l − 1 if x = tl−1 and n = l, otherwise. Since|N(x)| ≥ 2k − 1, |N(x) \ (A′ ∪ {sl, . . . , sk−1, tl, . . . , tk−1}| ≥ 2k − 1 − m −2(k−l) = 2l−m−1 ≥ l−m. Consequently, there is a subset B ⊆ N(x)\(A′∪{sl, . . . , sk−1, tl, . . . , tk−1} such that |B| ≥ l − m. Let B = {ym, . . . , yn−1},and let B′ = A′ ∪ B. It follows from the induction hypothesis, that G − xcontains pairwise disjoint paths Q0, . . . , Qn−1, Pl, . . . , Pk−1 such that Qi isthe trivial path consisting of ti for i ∈ {0, . . . ,m − 1}, Qi links yi to tifor i ∈ {m, . . . , n − 1}, and Pi links si to ti for i ∈ {l, . . . , k − 1}. Fori ∈ {0, . . . , n − 1} let Pi be the path obtained from Qi by adding the edge{yi, x}. If tl−1 = x, let Pl−1 be the trivial path consisting of x. Obviously,the paths P0, . . . , Pk−1 form the desired L-linkage in G.Next, we prove that (b) implies (c). Let G be k-linked and let T =(a0, . . . , ak−1) be an admissible sequence together with an admissible orien-tation of the edges. A vertex in V (T ) is said to be isolated if it is not incidentwith any edge in E(T ). Let M denote the set of all isolated vertices in V (T ),and let T ′ = (ai0 , . . . , air−1) be the subsequence of T obtained by deletingall elements ai ∈ V (T ) \ M . For e ∈ E(T ) let s(e) and t(e) denote thehead and the tail of e, respectively, and set s(x) = t(x) = x for all x ∈ M .Let L = {(s0, t0), . . . , (sr−1, tr−1)} where sj = s(aij ) for 0 ≤ j ≤ r − 1,tj = t(aij+1) for 0 ≤ j ≤ r − 2, and tr−1 = t(ai0). Since G is k-linked thereis an L-linkage, and it is not hard to see that the union of an L-linkage andE(T ) forms the desired cycle.Eventually, we prove that (c) implies (a) It is clear that every strongly k-vertex-edge-ordered graph is connected. Let G be a connected chordal graphon at least 2k vertices that is not (2k − 1)-connected. G has a minimalseparator S ⊆ V (G) with r = |S| ≤ 2k − 2. Let G1, G2 be two distinctcomponents of G − S. Since G is chordal, the subgraph H of G inducedby S is complete. Let Q = v1, . . . , vr be a Hamiltonian path of H, and letOrdered and Linked Chordal Graphs 373u1 and u2 be vertices of G1 and G2, respectively, such that u1 is adjacentto v1 and u2 is adjacent to vr in G. For 1 ≤ i ≤ br−12c, let ei denotethe oriented edge (v2i, v2i+1). Furthermore, let e0 and eb r−12c+1 denote theoriented edges (u1, v1) and (vr, u2), respectively. It is not hard to see, thatG does not contain a cycle that can be traversed such that e0, . . . , eb r−12c+1are encountered in the given order and every edge is traversed accordingto its orientation. Since b r−12c + 1 ≤ k, this shows that G is not stronglyk-vertex-edge-ordered. Hence every strongly k-vertex-edge-ordered chordalgraph is (2k − 1)-connected.References[1] B. Bolloba´s and A. Thomason, Highly linked graphs, Combinatorica 16 (1996)313–320.[2] R. Diestel, Graph Theory, Graduate Texts in Mathematics 173 (Springer,2000).[3] G.A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25(1961) 71–76.[4] R.J. Faudree, Survey on results on k-ordered graphs, Discrete Math. 229 (2001)73–87.[5] H.A. Jung, Eine veralgemeinerung des n-fachen zusammenhangs fu¨r graphen,Math. Ann. 187 (1970) 95–103.[6] D.G. Larman and P. Mani, On the existence of certain configurations withingraphs and the 1-skeleton of polytopes, Proc. London Math. Soc. 20 (1970)144–160.[7] L. Ng and M. Schultz, k-ordered Hamiltonian graphs, J. Graph Theory 24(1997) 45–57.[8] R. Thomas and P. Wollan, An improved linear edge bound for graph linkages,to appear in European J. Comb.Received 20 September 2007Revised 1 April 2008Accepted 2 April 2008Powered by TCPDF (www.tcpdf.org)

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Ordered and linked chordal graphs

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Ordered and linked chordal graphs

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