On small balanceable, strongly-balanceable and omnitonal graphs

Abstract

In Ramsey Theory for graphs we are given a graph G and we are required to find the least n 0 such that, for any n ≥ n 0 , any red/blue colouring of the edges of K n gives a subgraph G all of whose edges are blue or all are red. Here we shall be requiring that, for any red/blue colouring of the edges of K n , there must be a copy of G such that its edges are partitioned equally as red or blue (or the sizes of the colour classes differs by one in the case when G has an odd number of edges). This introduces the notion of balanceable graphs and the balance number of G which, if it exists, is the minimum integer bal(n, G) such that, for any red/blue colouring of E(K n ) with more than bal(n, G) edges of either colour, K n will contain a balanced coloured copy of G as described above. The strong balance number sbal(n, G) is analogously defined when G has an odd number of edges, but in this case we require that there are copies of G with both one more red edge and one more blue edge.These parameters were introduced by Caro, Hansberg and Montejano. These authors also introduce the more general omnitonal number ot(n, G) which requires copies of G containing a complete distribution of the number of red and blue edges over E(G).In this paper we shall catalogue bal(n, G), sbal(n, G) and ot(n, G) for all graphs G on at most four edges. We shall be using some of the key results of Caro et al. which we here reproduce in full, as well as some new results which we prove here. For example, we shall prove that the union of two bipartite graphs with the same number of edges is always balanceable.

Unavoidable chromatic patterns in 2‐colorings of the complete graph

“…After this, we focus our attention to the family of trees, particularly paths and stars (see Sections 3.4, 3.2, and 3.3, respectively). We shall mention here that other recent results in this flavor were obtained in [10].…”

On Zero-Sum Spanning Trees and Zero-Sum Connectivity

“…, G q = H such that, for every 1 i q − 1, G i ∼ = G and G i+1 is obtained from G i by an edge-replacement. The notion of amoebas was introduced in [11], further developed in [12,10] and also used in [13]. A major result in [12] states that a graph G is a global amoeba if and only if G ∪ K 1 is a local amoeba, meaning that the minimum possible n 0 as taken above is never larger than n + 1.…”

“…The appearance of new versions of zero-sum problems, where the range set is not Z k , but elements in Z, mostly {−1, 0, 1} or {−1, 1}, started in [13] (see also [5,6,7,9]), although an early paper about possible weights of spanning trees of the n-dimensional cube Q n under a {−1, 1}-colouring of its edges appeared in [18]. One of the first questions considered was: under what conditions does f : E(K n ) → {−1, 0, 1} force a zero-sum (over Z) spanning tree?…”

“…, G q = H such that, for every 1 ≤ i ≤ q − 1, G i ∼ = G and G i+1 is obtained from G i by an edge-replacement. The notion of amoebas was introduced in [5], further developed in [8] and also used in [9].…”

On the balanceability of some graph classes

“…Amoebas, a wide family of balanceable graphs with interesting interpolation properties were introduced in [5] and further developed in [6]. Caro, Lauri and Zarb [7] exhaustively studied the balanceability of graphs of at most four edges. For other references related to balanceability of graphs, see also [1,2,4].…”