Caterpillar arboricity of planar graphs

Research problems from the 18th British Combinatorial Conference

We prove that for every e>0 and positive integer r, there exists Δ0=Δ0(e) such that if Δ>Δ0 and n>n(Δ,e,r) then there exists a packing of K n with ⌊(n−1)/Δ⌋ graphs, each having maximum degree at most Δ and girth at least r, where at most en 2 edges are unpacked. This result is used to prove the following: Let f be an assignment of real numbers to the edges of a graph G. Let α(G,f) denote the maximum length of a monotone simple path of G with respect to f. Let α(G) be the minimum of α(G,f), ranging over all possible assignments. Now let αΔ be the maximum of α(G) ranging over all graphs with maximum degree at most Δ. We prove that Δ+1≥αΔ≥Δ(1−o(1)). This extends some results of Graham and Kleitman [6] and of Calderbank et al. [4] who considered α(K n ).

/pdf/large-monotone-paths-in-graphs-with-bounded-degree-38kathd42d.pdf

Large Monotone Paths in Graphs with Bounded Degree

Altitude of regular graphs with girth at least five

How long a monotone path can one always find in any edge-ordering of the complete graph Kn? This appealing question was first asked by Chvatal and Komlos in 1971, and has since attracted the attention of many researchers, inspiring a variety of related problems. The prevailing conjecture is that one can always find a monotone path of linear length, but until now the best known lower bound was n^2/3−o(1). In this paper we almost close this gap, proving that any edge-ordering of the complete graph contains a monotone path of length n^1−o(1).

/pdf/nearly-linear-monotone-paths-in-edge-ordered-graphs-1xpz05zw7f.pdf

Nearly-linear monotone paths in edge-ordered graphs

http://research.haifa.ac.il/~raphy/papers/monotone.pdf

Monotone paths in edge-ordered sparse graphs

In this paper performance analysis of a parallel Levinson-type algorithm for Toeplitz matrices is given. A modified version of the parallel algorithm is presented to improve performance. The algorithm is implemented on a shared-memory MIMD (non-vector) machine. The derivation of the parallel algorithm is presented. The speedup limitation is investigated along with the optimal number of processors needed to a given Toeplitz matrix.

Practical aspects and experiences: Timing analysis of a parallel algorithm for Toeplitz matrices on a MIMD parallel machine

Abstract In this paper performance analysis of a parallel Levinson-type algorithm for Toeplitz matrices is given. A modified version of the parallel algorithm is presented to improve performance. The algorithm is implemented on a shared-memory MIMD (non-vector) machine. The derivation of the parallel algorithm is presented. The speedup limitation is investigated along with the optimal number of processors needed to a given Toeplitz matrix.

Timing Analysis of a Parallel Algorithm for Toeplitz Matrices on a MIMD Parallel Machine.

On broadcasting multiple messages in a d -dimensional grid

https://cs.tau.ac.il/~amir1/PS/applied.pdf

Efficient line broadcasting in a d -dimensional grid

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