The vertex-arboricity a(G) of a graph G is the minimum number of subsets into which the set of vertices of G can be partitioned so that each subset induces a forest. It is well known that a(G) ≤ 3 for any planar graph G, and that a(G) ≤ 2 for any planar graph G of diameter at most 2. The conjecture that every planar graph G without 3-cycles has a vertex partition (V1, V2) such that V1 is an ind...

We consider the problem of edge orientation, whose goal is to orient the edges of an undirected dynamic graph with n vertices such that vertex out-degrees are bounded, typically by a function of the graph’s arboricity. Our main result is to show that an O(βα)-orientation can be maintained in O( lg(n/(βα)) β ) amortized edge insertion time and O(βα) worst-case edge deletion time, for any β ≥ 1, ...

Let H be a graph on h vertices, and let G be a graph on n vertices. An H-factor of G is a spanning subgraph of G consisting of n/h vertex disjoint copies of H. The fractional arboricity of H is a(H) = max{ |E ′| |V ′|−1}, where the maximum is taken over all subgraphs (V ′, E′) of H with |V ′| > 1. Let δ(H) denote the minimum degree of a vertex of H. It is shown that if δ(H) < a(H) then n−1/a(H)...

Wu, Zhang and Li [4] conjectured that the set of vertices of any simple graph G can be equitably partitioned into ⌈(∆(G) + 1)/2⌉ subsets so that each of them induces a forest of G. In this note, we prove this conjecture for graphs G with ∆(G) ≥ |G|/2.

For two graphs G and H , let the mixed anti-Ramsey numbers, maxR(n; G, H), (minR(n; G, H)) be the maximum (minimum) number of colors used in an edge-coloring of a complete graph with n vertices having no monochromatic subgraph isomorphic to G and no totally multicolored (rainbow) subgraph isomorphic to H . These two numbers generalize the classical anti-Ramsey and Ramsey numbers, respectively. ...

A graph is k-degenerate if any induced subgraph has a vertex of degree at most k. In this paper we prove new algorithms finding cliques and similar structures in these graphs. We design linear time Fixed-Parameter Tractable algorithms for induced and non induced bicliques. We prove an algorithm listing all maximal bicliques in time O(k(n−k)2), improving the result of [D. Eppstein, Arboricity an...

Abstract We consider arc colourings of oriented graphs such that for each vertex the colours all out-arcs incident with and in-arcs form intervals. prove existence a colouring is an NP-complete problem. give solution problem r -regular graphs, transitive tournaments, small maximum degree, order some other classes graphs. state conjecture graph there exists consecutive colourable orientation con...

For a graph theoretic parameter f, an integer m and a graph H, the mixed Ramsey number r(f‘; m; H) is defined as the least positive integer p such that if G is any graph of order p, then either ,f( G) >m or ?? contains a subgraph isomorphic to H Let /j denote vertex linear arboricity and let H be any connected graph of order n. In this note we show that c(p; m; H ) = 1 + (n + n,,(g) 2)(m l), wh...

We prove that a 4-connected K 4;4-minor free graph on n vertices has at most 4n ? 8 edges and we use this result to show that every K 4;4-minor free graph has vertex-arboricity at most 4. This improves the case (n; m) = (7; 3) of the following conjecture of Woodall: the vertexset of a graph without a K n-minor and without a K b n+1 2 c;d n+1 2 e-minor can be partitioned in n ? m + 1 subgraphs w...