Coverings by Few Monochromatic Pieces: A Transition Between Two Ramsey Problems

The typical problem in (generalized) Ramsey theory is to find the order of the largest monochromatic member of a family {Mathematical expression} (for example matchings, paths, cycles, connected subgraphs) that must be present in any edge coloring of a complete graph Kn with t colors. Another area is to find the minimum number of monochromatic members of {Mathematical expression} that partition or cover the vertex set of every edge colored complete graph. Here we propose a problem that connects these areas: for a fixed positive integers s ≤ t, at least how many vertices can be covered by the vertices of no more than s monochromatic members of {Mathematical expression} in every edge coloring of Kn with t colors. Several problems and conjectures are presented, among them a possible extension of a well-known result of Cockayne and Lorimer on monochromatic matchings for which we prove an initial step: every t-coloring of Kn contains a (t - 1)-colored matching of size k provided that {Mathematical expression} © 2013 Springer Japan

Counting irregular multigraphs

AbstractGagliardi et al. (1996, unpublished manuscript) defined an irregular multigraph to be a loopless multigraph with degree sequence n, n − 1,…, 1, and they posed the problem of determining the number of different irregular multigraphs fn on n vertices. In Gagliardi et al. (1996) they showed that if n ≡ 0 or 3 (mod 4) then fn > n − 1. In this note our aim is to show that there are constants 1 < c1 < c2 and n0 > 0 such that if n ⩾ n0 and n ≡ 0 or 3 (mod 4) then (c1)n2 < fn < (c2)n2. Indeed, we show that c1 = 1.19 and c2 = 1.65 can be chosen

On a question of Gowers concerning isosceles right-angle triangles

We give a simple quantitative proof that for every natural number p _&gt; 3 and real number 5 &gt; 0, there is a natural number No = No(p, 5) such that for N _&gt; No, every set of at least 5N 2 points of [N] 2 contains a set ofp points that determine at least p - [log 2 p] isosceles right-angle triangles; i.e. triples in the form ((a, b), (a +a,b),(a,b+ a)}

Distributing vertices along a Hamiltonian cycle in Dirac graphs

AbstractA graph G on n vertices is called a Dirac graph if it has a minimum degree of at least n/2. The distance distG(u,v) is defined as the number of edges in a shortest path of G joining u and v. In this paper we show that in a Dirac graph G, for every small enough subset S of the vertices, we can distribute the vertices of S along a Hamiltonian cycle C of G in such a way that all but two pairs of subsequent vertices of S have prescribed distances (apart from a difference of at most 1) along C. More precisely we show the following. There are ω,n0>0 such that if G is a Dirac graph on n≥n0 vertices, d is an arbitrary integer with 3≤d≤ωn/2 and S is an arbitrary subset of the vertices of G with 2≤|S|=k≤ωn/d, then for every sequence di of integers with 3≤di≤d,1≤i≤k−1, there is a Hamiltonian cycle C of G and an ordering of the vertices of S, a1,a2,…,ak, such that the vertices of S are visited in this order on C and we have |distC(ai,ai+1)−di|≤1,for all but one1≤i≤k−1

Distributing vertices along a Hamiltonian cycle in Dirac graphs

A graph G on n vertices is called a Dirac graph if it has minimum degree at least n=2. The distance dist G (u; v) is de ned as the number of edges in a shortest subpath of G joining u and v. In this paper we show that in a Dirac graph G, for every small enough subset A of the vertices, we can distribute the vertices of A along a Hamiltonian cycle C of G in such a way that all but two pairs of subsequent vertices of A have prescribed distances (apart from a dierence of at most 1) along C. More precisely we show the following. There are &quot;; n 0 &gt; 0 such that if G is a Dirac graph on n n 0 vertices, d is an arbitrary integer with 3 d &quot;n=2 and A is an arbitrary subset of the vertices of G with 2 jAj = k &quot;n=d, then for every sequence d i of integers with 3 d i d; 1 i k 1, there is a Hamiltonian cycle C of G and an ordering of the vertices of A, a 1 ; a 2 ; : : : ; a k , such that the vertices of A are visited in this order on C and we have

On an anti-Ramsey problem of Burr, Erdös, Graham and T. Sós

Given a graph L, in this paper we investigate the anti-Ramsey number S (n; e; L), de ned to be the minimum number of colors needed to edge-color some graph G(n; e) with n vertices and e edges so that in every copy of L in G all edges have dierent colors. We call such a copy of L totally multicolored (TMC)

On Edge Colorings With At Least Q Colors in Every Subset of P Vertices

For fixed integers p and q, an edge coloring of K n is called a (p; q)-coloring if the edges of K n in every subset of p vertices are colored with at least q distinct colors. Let f(n; p; q) be the smallest number of colors needed for a (p; q)-coloring of K n . In [3] Erdos and Gyárfás studied this function if p and q are fixed and n tends to infinity. They determined for every p the smallest q (= \Gamma p 2 \Delta \Gamma p + 3) for which f(n; p; q) is linear in n and the smallest q for which f(n; p; q) is quadratic in n. They raised the question whether perhaps this is the only q value which results in a linear f(n; p; q). In this paper we study the behavior of f(n; p; q) between the linear and quadratic orders of magnitude. In particular we show that that we can have at most log p values of q which give a linear f(n; p; q)