Paths, cycles and wheels in graphs without antitriangles

We investigate paths, cycles and wheels in graphs with independence number of at most 2, in particular we prove theorems characterizing ali such graphs which are hamiltonian. Ramsey numbers of the form R (G, K 3)' for G being a path, a cycle or a wheel, are known to be 2n (G) 1, except for some small cases. In this paper we derive and count all critical graphs for these Ramsey numbers. 1. Notation and Previous Work For any graph F, V (F) and E (F) will denote the vertex and edge sets of the graph F, also let n (F) = I V (F) I and e (F) = IE (F) I. The graph F denotes the complement of F. A graph F will be called a (G, H)-good graph, if F does not contain G and F does not contain H. Any (G, H)-good graph on n vertices will be called a (G,H,n)-good graph. The Ramsey number R(G,H) is defined as the smallest integer n such that no (G, H ,n )-good graph exists. Any graph is called a critical graph for the Ramsey number R (G ,H) if it is (G, H ,R (G ,H) -I)-good. When the graph F is fixed, then for any vertex XE V(F), Gx and Hx will denote the graphs induced by the neighbors of the vertex x or by all the vertices disconnected from x, respectively. Pi is a path on i vertices, Ci is a cycle of length i, and Wi is a wheel with i-I spokes, i.e. a graph formed by some vertex x, called a hub· of the wheel, connected to all vertices of some cycle Ci 1, called a rim. 2Ki is the graph fonned by two vertex disjoint copies of K j • For notational convenience we define C j =Ki for 1 -:::;, i -:::;, 2. In this paper most of the graphs considered are (T, K 3, n )-good for T being a path, a cycle, or a wheel. It is easy to see that if F is any (T, K 3, n )-good graph and x is a vertex in V (F) of degree deg (x) = d, then: (a) if T =C i +1 then Gx is a (Pi,K3,d)-good graph, (b) if T Wi+1 then G.>; is a (Ci ,K3,d)-good graph, and Hx is a complete graph Kn -d -1' Observe that any graph without independent sets of size three and with more than one component is a vertex disjoint union of two cliques. We also note that the whole contents of this paper can be seen as a study of paths, cycles and wheels in the complements of triangle free graphs. The value of the Ramsey number R (Pi, K 3) = 2i 1 is a consequence of a well known theorem by Chvatal [3]. An interesting general related result in [2J Says that R (G, K 3) = 2i -1 for any connected graph G of order i ;?: 4 with at most (l7i + 1)115 edges, which obviously applies to the cases of paths and cycles, but not wheels. Burr and Erdos (1] showed that R (Wi' K 3) = 2i 1 for all i :2: 6, and the tables by Clancy [4] include the special value of R (W 5, K 3) = 11. McKay and Faudree [5] generated and counted by computer all of the critical graphs for the Ramsey numbers R (Wj , K 3) for all j ::; 11, and our proofs confirm their results. In two recent papers Sidorenko studied the general case: in (7] he showed that for any graph G without isolated vertices we have R (G , K 3}::; 2e (G) + 1, which improved on his previous result in [6], where he also formulated an interesting conjecture that for any graph G there is a general bound R(G,K3)::;n(G)+e(G). Sidorenko's result in [7] proves Harary's conjecture fom1Ulated in 1980. We derive a characterization of all hamiltonian graphs with independence number at most 2. For T being any of Pi, Ci or Wi we will describe and count all of the critical graphs for the Ramsey numbers R (T ,K 3), in particular we will prove that almost all such critical graphs must contain 2Ki1• The latter will also give alternate proofs of previously known results that for the same possible T's and for all i ;?: 1 we have R (T, K 3) 2i 1, except some small cases listed in Theorems 3 and 5. We include these alternate proofs, so the results of this paper are self contained.