Unavoidable chromatic patterns in 2-colorings of the complete graph

Abstract

Let G be a graph with e(G) edges. We say that G is omnitonal if for every sufficiently large n there exists a minimum integer ot(n, G) such that the following holds true: For any 2-coloring f : E(K n ) → {red, blue} with more than ot(n, G) edges from each color, and for any pair of non-negative integers r and b with r + b = e(G), there is a copy of G in K n with exactly r red edges and b blue edges. We give a structural characterization of omnitonal graphs from which we deduce that omnitonal graphs are, in particular, bipartite graphs, and prove further that, for an omnitonal graph, where m = m(G) depends only on G. We also present a class of graphs for which ot(n, G) = ex(n, G), the celebrated Turán numbers. Many more results and problems of similar flavor are presented.

Graphs isomorphisms under edge-replacements and the family of amoebas

“…Graphs called amoebas first appeared in [3] where certain Ramsey-Turán extremal problems were considered, which dealt with the existence of a given graph with a prescribed color pattern in 2-edge-colorings of the complete graph. More precisely, amoebas arose from the search of a graph family with certain interpolation properties that are suitable for the techniques to show balanceability or omnitonal properties, see [2,3] for a deeper insight into this matter. For the interested reader, we refer to [1,6,7,9,10,12,14] for more literature related to interpolation techniques in graphs.…”

“…Note that for the concept of global amoeba, which is the one considered already in the literature [3], we can maintain the image of a graph G embedded in a complete graph K N , with N = n + t much larger than |V (G)| = n, traveling via feasible edge replacements from any given copy of it to any other one.…”

On the balanceability of some graph classes

“…However, it is also possible to look for other kinds of ordered substructures. In particular, Caro, Hansberg and Montejano [1] introduced the notion of balanceability, which looks for balanced copies of a graph in 2-colorings of the edges of K n .…”

“…Observation 2. Note that the concept of balanceability can be extended to graphs with an odd number of edges, as explained in [1]. In this case, we would be looking for a copy where the number of edges in each color class differ by 1.…”

“…Given a graph G and a partition (X, Y ) of its vertices, we denote by e(X, Y ) the number of edges with one endpoint in X and the other in Y ; given a graph G and a subset W of its vertices, we denote by e(G[W ]) the number of edges in the subgraph of G induced by the vertices in W . In their paper, Caro, Hansberg and Montejano proved the following characterization of balanceable graphs: Theorem 3 ( [1]). A graph G(V, E) with an even number of edges is balanceable if and only if G has both a partition V = X ∪ Y and a set of vertices W ⊆ V such that e(X, Y ) = e(G[W ]) = |E| 2 .…”