Tomasz Dzido

MULTICOLOR RAMSEY NUMBERS FOR PATHS AND CYCLES

For given graphs G1;G2;:::;Gk;k ‚ 2, the multicolor Ramsey number R(G1;G2;:::;Gk) is the smallest integer n such that if we arbi- trarily color the edges of the complete graph on n vertices with k colors, then it is always a monochromatic copy of some Gi, for 1 • ik. We give a lower bound for k-color Ramsey number R(Cm;Cm;:::;Cm), where m ‚ 8 is even and Cm is the cycle on m vertices. In addition, we provide exact values for Ramsey numbers R(P3;Cm;Cp), where P3 is the path on 3 vertices, and several values for R(Pl;Pm;Cp), where l;m;p ‚ 2. In this paper we present new results in this fleld as well as some interesting conjectures.

On a local similarity of graphs

We say that two graphs G and H , having the same number of vertices n , are k -similar if they contain a common induced subgraph of order k . We will consider the following question: how large does n need to be to ensure at least one k -similar pair in any family of l graphs on n vertices? We will present various lower and upper bounds on n . In particular, we will prove that for l = 3 , n equals the Ramsey number R ( k , k ) . Last but not least we will determine the exact values of n for k = 3 , k = 4 and all l .

Altitude of wheels and wheel-like graphs

An edge-ordering of a graph G=(V, E) is a one-to-one mapping f:E(G)→{1, 2, ..., |E(G)|}. A path of length k in G is called a (k, f)-ascent if f increases along the successive edges forming the path. The altitude α(G) of G is the greatest integer k such that for all edge-orderings f, G has a (k, f)-ascent.In our paper we give exact values of α(G) for all helms and wheels. Furthermore, we use our result to obtain altitude for graphs that are subgraphs of helms.

Turán numbers for odd wheels

Abstract The Turan number ex ( n , G ) is the maximum number of edges in any n -vertex graph that does not contain a subgraph isomorphic to G . A wheel W n is a graph on n vertices obtained from a C n − 1 by adding one vertex w and making w adjacent to all vertices of the C n − 1 . We obtain two exact values for small wheels: ex ( n , W 5 ) = ⌊ n 2 4 + n 2 ⌋ , ex ( n , W 7 ) = ⌊ n 2 4 + n 2 + 1 ⌋ . Given that ex ( n , W 6 ) is already known, this paper completes the spectrum for all wheels up to 7 vertices. In addition, we present the construction which gives us the lower bound ex ( n , W 2 k + 1 ) > ⌊ n 2 4 ⌋ + ⌊ n 2 ⌋ in general case.

On some three-color Ramsey numbers for paths

For graphs G 1 , G 2 , G 3 , the three-color Ramsey number R ( G 1 , G 2 , G 3 ) is the smallest integer n such that if we arbitrarily color the edges of the complete graph of order n with 3 colors, then it contains a monochromatic copy of G i in color i , for some 1 ? i ? 3 .First, we prove that the conjectured equality R ( C 2 n , C 2 n , C 2 n ) = 4 n , if true, implies that R ( P 2 n + 1 , P 2 n + 1 , P 2 n + 1 ) = 4 n + 1 for all n ? 3 . We also obtain two new exact values R ( P 8 , P 8 , P 8 ) = 14 and R ( P 9 , P 9 , P 9 ) = 17 , furthermore we do so without help of computer algorithms. Our results agree with a formula R ( P n , P n , P n ) = 2 n - 2 + ( n mod 2 ) which was proved for sufficiently large n by Gyarfas, Ruszinko, Sarkozy, and Szemeredi (2007). This provides more evidence for the conjecture that the latter holds for all n ? 1 .

The upper domination Ramsey number u(4,4)

The upper domination Ramsey number u(m,n) is the smallest integer p such that every 2-coloring of the edges of Kp with color red and blue, Γ(B) ≥ m or Γ(R) ≥ n, where B and R is the subgraph of Kp induced by blue and red edges, respectively; Γ(G) is the maximum cardinality of a minimal dominating set of a graph G. In this paper, we show that u(4, 4) ≤ 15.