Let G = (V,E) be an n-vertex graph and Md a d-vertex graph, for some constant d. Is Md a subgraph of G? We consider this problem in a model where all n processes are connected to all other processes, and each message contains up to O(log n) bits. A simple deterministic algorithm that requires O(n(d−2)/d/ log n) communication rounds is presented. For the special case that Md is a triangle, we pr...

A tour in a graph is a connected walk that visits every vertex at least once, and returns to the starting vertex. Vishnoi (2012) proved that every connected d-regular graph with n vertices has a tour of length at most (1 + o(1))n, where the o(1) term (slowly) tends to 0 as d grows. His proof is based on van-derWarden’s conjecture (proved independently by Egorychev (1981) and by Falikman (1981))...

Let G = (V,E) be an n-vertex graph and Md a d-vertex graph, for some constant d. Is Md a subgraph of G? We consider this problem in a model where all n processes are connected to all other processes, and each message contains up to O(logn) bits. A simple deterministic algorithm that requires O(n(d−2)/d/ logn) communication rounds is presented. For the special case that Md is a triangle, we pres...

The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edges of G. In this paper, it is proved that if G is a planar graph with ∆(G) ≥ 5 and without 4-cycles, then la(G) = ⌈∆(G) 2 ⌉. Moreover, the bound that ∆(G)≥ 5 is sharp.

The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edges of G. An and Wu introduce the notion of list linear arboricity lla(G) of a graph G and conjecture that lla(G) = la(G) for any graph G. We confirm that this conjecture is true for any planar graph having ∆ > 13, or for any planar graph with ∆ > 7 and without i-cycles for some i ∈ {3, 4, 5}....

For a loopless multigraph G, the fractional arboricity Arb(G) is the maximum of |E(H)| |V (H)|−1 over all subgraphs H with at least two vertices. Generalizing the NashWilliams Arboricity Theorem, the Nine Dragon Tree Conjecture asserts that if Arb(G) ≤ k + d k+d+1 , then G decomposes into k + 1 forests with one having maximum degree at most d. The conjecture was previously proved for d = k+1 an...

For a xed positive integer k, the linear k-arboricity lak(G) of a graph G is the minimum number ‘ such that the edge set E(G) can be partitioned into ‘ disjoint sets and that each induces a subgraph whose components are paths of lengths at most k. This paper studies linear k-arboricity from an algorithmic point of view. In particular, we present a linear-time algorithm to determine whether a tr...