New lower bound for multicolor Ramsey numbers for even cycles.
On some Ramsey and Turán-type numbers for paths and cycles.
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 arbitrarily color the edges of the complete graph on n vertices with k colors, then it is always a monochromatic copy of some , for 1 ≤ i ≤ k. 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...
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.
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.
Edit distance measure for graphs
In this paper, we investigate a measure of similarity of graphs similar to the Ramsey number. We present values and bounds for , the biggest number guaranteeing that there exist graphs on vertices, each two having edit distance at least . By edit distance of two graphs , we mean the number of edges needed to be added to or deleted from graph to obtain graph . This new extremal number is closely linked to the edit distance of graphs. Using probabilistic methods we show that is close...
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