Another Infinite Sequence of Dense Triangle-Free Graphs

The core is the unique homorphically minimal subgraph of a graph A triangle free graph with minimum degree n is called dense It was observed by many authors that dense triangle free graphs share strong structural properties and that the natural way to describe the structure of these graphs is in terms of graph homomorphisms One in nite sequence of cores of dense maximal triangle free graphs was known All graphs in this sequence are colourable Only two addi tional cores with chromatic number were known We show that the additional graphs are the initial terms of a second in nite sequence of cores Let G and H be graphs A homomorphism G H is a function V G V H mapping edges on edges i e vw E G implies v w E H Every graph G has a unique minimal subgraph G with G G which is called the core for an introduction and relevant literature see Note that the core of G has the same chromatic number as G A triangle free graph is maximal if for every pair of non adjacent vertices u v the addition of the edge uv creates a triangle Note that a triangle free FB Mathematik Informatik WE Freie Universit at Berlin Arnimallee Berlin Germany e mail brandt math fu berlin de University of Ljubljana Faculty of Mathematics and Physics and IMFM TCS Jad ranska Ljubljana Slovenia e mail Tomaz Pisanski fmf uni lj si graph of order n is maximal if and only if it has diameter Let u u be two non adjacent vertices of a maximal triangle free graph G Assume that there is a vertex v which is adjacent to u but not to u Since u and v are not adjacent they must have a common neighbour v Therefore u u for any homomorphism from G to a triangle free graph It follows that w w implies that the set of neighbours of w is the same as the set of neighbours of w Two vertices sharing the same neighbourhood have been called twins or similar or symmetric The twin relation is an equivalence relation on the vertex set of a graph where every equivalence class forms an independent set The identi cation of twins in any order eventually leads to the unique maximal twin free induced subgraph which coincides with the core if the graph is maximal triangle free Hence the core of a maximal triangle free graph G is obtained by successively identifying twins until no twins remain The reverse operation to twin identi cation is vertex duplication where at each step a new twin is added to a vertex The nal result depends only on the number of times a vertex is duplicated and not on the order of the duplications Hence the result is uniquely described if we assign positive integers to the vertices of the original graph each integer specifying the number of duplicates including the original vertex In case of a core the numbers represent the cardinalities of the equivalence classes of the twin relation A triangle free graph of order n with minimum degree n will be called dense Chen Jin and Koh characterized the dense maximal triangle free colourable graphs in terms of their cores see Brandt for a much simpler proof Only two chromatic cores of dense maximal triangle free were known one being the Gr otzsch graph detected by H aggkvist and the other being the graph which was found by Jin see Figure The objective of this note is to show that these two graphs are the initial terms of an in nite sequence of chromatic cores of dense maximal triangle free graphs In fact we show that for every p there exists a chromatic graph p on p vertices being the core of a dense triangle free graph Let us start with describing an in nite sequence k of colourable dense maximal triangle free cores Later these graphs will appear as building blocks in the new sequence p Figure The Gr otzsch graph and the Jin graph For k let k denote the Cayley graph over Z k with respect to the set of generators fi k i k g These graphs are complements of cycle powers de ned as follows Let K and for k k C k k i e k is the complement of the k st power of the k cycle As usual the kth power G of a graph G is the graph on V G where v and w are adjacent if and only if their distance in G is at most k Clearly C the cycle and is the M obius ladder on vertices The graph k can also be described as the Cayley graph over Z k with respect to the set of generators f i i kg We will make use of this representation by assuming that k has vertex set fw w k g where wiwj E k i ji jj mod This sequence of graphs was probably rst discovered by Andrasfai in and has been rediscovered several times throughout the years In Pach observed that the triangle free graphs where every independent set of vertices is contained in the neighbourhood of a vertex are precisely those which can be obtained from a graph k by consecutively duplicating vertices Pach s result was recently rediscovered by Brouwer giving a much shorter proof In it was observed that these graphs can be alternatively characterized as those maximal triangle free graphs which do not contain an induced cycle Note that the duplication of a vertex does not change the core of the Figure is composed of two parts The graph on vertices on the left to which is attached on the right graph since the vertex and its twin can be identi ed by a homomorphism Generalizing a result of Jin Chen Jin and Koh proved that the core of every colourable dense maximal triangle free graph is a graph k So the question arose whether it is possible to characterize the cores of chromatic dense triangle free graphs as well It was proven by Chen Jin and Koh that every such graph contains the Gr otzsch graph as an induced subgraph see also for a simple proof of these results Using the computer program Vega we computed further graphs which are cores of dense maximal triangle free chromatic graphs and we observed that they belong to an in nite sequence of such graphs which we will call due to their origin Vega graphs For every p there exists a Vega graph p with p vertices which is obtained in the following way For k and p k take a cycle C v v v an edge u u and a copy of k with vertex set fw w w k g Join u to v v v u to v v v and w i j to v j and v j for i k and j The resulting graph is k see Figure for the case k In order to obtain k delete the vertex w from k and to obtain k delete the vertex u from k Now we can state our main result Theorem For every p there is a triangle free graph with chromatic number of order p k which is the core of a triangle free k regular graph on n k vertices Proof We claim that the Vega graphs p p have the indicated property First we show that p is its own core which follows from the fact that p is maximal triangle free and the neighbourhood of no vertex is contained in the neighbourhood of another vertex This can be derived from the fact that the graphs k do have this property combined with a simple case analysis In particular p is the core of every graph G which is homomorphic with p and which contains p as a subgraph It is left to show that for every p there is a regular dense maximal triangle free graph whose core is p We do this by duplicating vertices according to an appropriate weight assignment Let us start assigning the weights to k and then modify the weights for the cases p k and p k For p k assign weight to each vertex ui and to the vertices w and w k weight to the vertices wi for i k weight k to v and v and weight k to v v v v The result by duplicating the vertices the right number of times is a k regular graph of order k For p k where the vertex w is deleted leave the labels unchanged except for w and w k which both get weight and v and v which both get weight k The result is a k regular graph of order k Finally the case p k Here u is additionally deleted We leave the labels unchanged except for u which gets label v and v which get label k and v which gets label k The result is a k regular graph of order k Finally we would like to know whether there are further cores of dense maximal triangle free graphs If not then the answer to the following problem was a rmative note that every graph k is a subgraph of k Problem Is every triangle free graph with minimum degree n ho momorphic with a graph p As a consequence this would imply that every triangle free graph with minimum degree n is colourable in contrast to a conjecture of Jin saying that there are graphs of arbitrarily large chromatic number with this property