Forbidden triples and traceability: a characterization

Given a connected graph G, a family F of connected graphs is called a forbidden family if no induced subgraph of G is isomorphic to any graph in F. If this is the case, G is said to be F-free. In earlier papers the authors identi ed four distinct families of triples of subgraphs that imply traceability when they are forbidden in su ciently large graphs. In this paper the authors introduce a fth family and show these are all such families. c © 1999 Elsevier Science B.V. All rights reserved. 1. Background and notation The graphs discussed here are simple graphs. For terms not de ned here, see [3]. Let G be a connected graph and let F be a family of connected graphs. We say that F is a family of forbidden subgraphs (or a forbidden family) if no induced subgraph of G is isomorphic to any graph in F. If this is the case, G is said to be F-free. If F consists of a single graph, say H , we say that G is H -free. A graph is said to be traceable if it contains a path that spans the vertex set. In two previous papers [4,5] four distinct families of triples of subgraphs were shown to imply traceability when forbidden in su ciently large graphs. The families are as follows (refer to Fig. 1 for the graphs themselves): 1. {K1;m; Yl; Z1} (m¿4; l¿4). 2. {K1;m; P4; Vr} (m¿4; r¿3). 3. {K1;3; Er; Z2} (r¿4). 4. {K1;m; Pl; Qr} (m¿4; l¿5; r¿3). ∗ Corresponding author. Tel.: +1-828-262-2374; fax: +1-828-265-8617. E-mail address: [email protected] (J.M. Harris) 1 Research supported in part by O.N.R. Grant N00014-97-1-0499. 0012-365X/99/$ see front matter c © 1999 Elsevier Science B.V. All rights reserved. PII: S0012 -365X(99)00021 -7 102 R.J. Gould, J.M. Harris / Discrete Mathematics 203 (1999) 101–120 Fig. 1. Graphs involved in forbidden triples. Characterizations have been discovered for all the single graphs and all the pairs of graphs that imply traceability when forbidden in connected graphs (see [2]). It should be noted that if any of these graphs (the single or the pairs) are contained in a triple T= {A; B; C}, then certainly a connected graph that is T-free will be traceable. The single and the pairs are described in Section 3 of this paper. In Section 2 we identify an additional family, {K1;3; Qr; Nk}, that enjoys the property of implying traceability in su ciently large graphs. In Section 3 we show that this family, along with the previous four, are the only nontrivial families of triples do this (that is, the only families not containing the single graph or one of the pairs mentioned above). Regarding notation, given two vertices v and w of a graph G, we let dG(v; w) denote the distance (the length of a shortest path) in G from v to w. If A is a subset of the vertices of G, we let 〈A〉 denote the subgraph of G induced by A. Also, given a vertex v, we let NA(v) denote the set of vertices in A that are adjacent to v. Finally, in a graph G, suppose we have internally disjoint paths P1 : a1; a2; : : : ; ai and P2 : b1; b2; : : : ; bj. If the edge aib1 exists, then the path P in G described by P : a1; : : : ; ai; b1; : : : ; bj will be denoted as [a1; ai]P1 ; [b1; bj]P2 . In a similar fashion, if ai = b1, then the notation given by [a1; ai]P1 ; (b1; bj]P2 will represent the path S in G given by S : a1; : : : ; ai; b2; : : : ; bj. R.J. Gould, J.M. Harris / Discrete Mathematics 203 (1999) 101–120 103 2. The family: {K1;3; Qr; Nk}(r¿4; k¿2) We begin this section by stating a result from Sumner (see [6, p. 142]) that we will use later. Note that (G) represents the connectivity of G. Theorem A (Sumner [6]). If G is a claw-free graph of order n; and if (G)¿n=4; then G is hamiltonian. Theorem 2.1. Let r¿4 and k¿2 be xed integers. Let G be a connected graph of order n that is {K1;3; Qr; Nk}-free. If n is su ciently large; then G is traceable. Proof. Let T be a minimum cut set of G, let v∈T , and let S=T \{v}. (It is possible that S = ∅.) We know that 〈V (G) \ T 〉 is either disconnected or a single vertex. If 〈V (G)\T 〉 is a single vertex, then |T |= |V (G)|−1, and hence G is a complete graph, and is certainly traceable. Therefore, assume that 〈V (G) \ T 〉 is disconnected. Since T is minimum, it must be that 〈V (G) \ S〉 is 1-connected and has v as a cut vertex. Now, if 〈V (G) \ T 〉 has more than two components, then there exist vertices a; b; c∈N (v) that are pairwise nonadjacent, and then we will have a claw: 〈{a; b; c; v}〉. Thus, 〈V (G)− T 〉 must have exactly two components, say A and B. We partition the vertices of A and B as follows. For i=1; 2; : : :, de ne Ai={u∈V (A) : d(u; v) = i} and Bi = {u∈V (B) : d(u; v) = i}. Further, de ne A0 = B0 = {v}. Note that since G is nite, there exists an integer l¿1 such that Al 6= ∅ and Ai = ∅ for i¿ l. Also, there must exist an integer m¿1 such that Bm 6= ∅ and Bi = ∅ for i¿m. We now make several Notes, each of which is easily veri ed: Note (a): Each vertex of S is adjacent to at least one vertex of A and to at least one vertex of B. Note (b): No vertex of A is adjacent to any vertex of B. Note (c): (i) N (Ai) ∩ Aj = ∅ for each i∈ 1; : : : ; l and for each j 6= i− 1; i; i+ 1; (ii) N (Bi) ∩ Bj = ∅ for each i∈ 1; : : : ; m and for each j 6= i − 1; i; i + 1. Note (d): For i¿1, if x∈Ai (resp. Bi), then x is adjacent to some vertex of Ai−1 (resp. Bi−1). Note (e): If x and y are nonadjacent vertices of Ai (resp. Bj), then x and y have no common neighbors in Ai−1 (resp. Bj−1). Note (f): The subgraphs 〈A1〉 and 〈B1〉 are complete. Note (g): If x is a vertex of Ai, then there exists an induced path P: x; ai−1; ai−2; : : : ; a1; v where x and v are the endpoints and aj ∈Aj for j = 1; 2; : : : ; i − 1. We make a de nition: given i∈{1; : : : ; l − 1}, some vertices of Ai are adjacent to vertices of Ai+1, while some vertices may not be. That is, some vertices of Ai “continue on” to Ai+1, and some do not continue. We will call a vertex x∈Ai a continuer if it is adjacent to some vertex of Ai+1. Otherwise, we call x a noncontinuer. The terms continuer and noncontinuer will have similar meanings in B. Note (h): Each of A0; A1; A2; : : : ; Al−1; B1; B2; : : : ; Bm−1 contains at least one continuer. 104 R.J. Gould, J.M. Harris / Discrete Mathematics 203 (1999) 101–120 Claim 2.1. For each i∈{1; 2; : : : ; l}; |Ai|¡ (r − 1)i ; and for each j∈{1; 2; : : : ; m}; |Bj|¡ (r − 1) j. Proof. We will prove the bound on |Ai| by induction. The argument for |Bj| is almost identical. From Note (f) above we know that 〈A1〉 is complete. If we suppose that |A1|¿r−1, and we let b1 be a vertex of B1, then we see that 〈A1∪{v}∪{b1}〉 contains an induced Qr . Thus, |A1|¡r − 1. Now, suppose the claim is true for Ai−1 where i¿2. Let ai−1 be a vertex of Ai−1, let ai−2 ∈Ai−2 be a neighbor of ai−1, and consider the vertices of NAi(ai−1). If vertices ai; ai ∈NAi(ai−1) are nonadjacent, then 〈{ai; ai ; ai−1; ai−2}〉 is an induced K1;3. Therefore, ai and ai must be adjacent, and we can then conclude that 〈NAi(ai−1)〉 must be complete. Thus, if |NAi(ai−1)|¿r−1, we again have a subgraph (〈NAi(ai−1)∪ {ai−1} ∪ {ai−2}〉) which contains an induced Qr . Hence |NAi(ai−1)|¡r − 1. Thus we have that |Ai|6 ∣∣∣∣∣ ⋃ x ∈ Ai−1 NAi(x) ∣∣∣∣∣¡ (r − 1)(r − 1)i−1 = (r − 1) ; and the claim is proved. Given the integers r and k, we let