On the connectivity and restricted edge-connectivity of 3-arc graphs

A 3−arc of a graph G is a 4-tuple (y, a, b, x) of vertices such that both (y, a, b) and (a, b, x) are paths of length two in G. Let ←→G denote the symmetric digraph of a graph G. The 3-arc graph X(G) of a given graph G is defined to have vertices the arcs of ←→G. Two vertices (ay), (bx) are adjacent in X(G) if and only if (y, a, b, x) is a 3-arc of G. The purpose of this work is to study the edge-connectivity and restricted edge-connectivity of 3-arc graphs.We prove that the 3-arc graph X(G) of every connected graph G of minimum degree δ(G) ≥ 3 has edge-connectivity λ(X(G)) ≥ (δ(G) − 1)2; and restricted edge- connectivity λ(2)(X(G)) ≥ 2(δ(G) − 1)2 − 2 if κ(G) ≥ 2. We also provide examples showing that all these bounds are sharp.Peer Reviewe

A Study on Graph Coloring and Digraph Connectivity

This dissertation focuses on coloring problems in graphs and connectivity problems in digraphs. We obtain the following advances in both directions.;1. Results in graph coloring. For integers k,r \u3e 0, a (k,r)-coloring of a graph G is a proper coloring on the vertices of G with k colors such that every vertex v of degree d( v) is adjacent to vertices with at least min{lcub}d( v),r{rcub} different colors. The r-hued chromatic number, denoted by chir(G ), is the smallest integer k for which a graph G has a (k,r)-coloring.;For a k-list assignment L to vertices of a graph G, a linear (L,r)-coloring of a graph G is a coloring c of the vertices of G such that for every vertex v of degree d(v), c(v)∈ L(v), the number of colors used by the neighbors of v is at least min{lcub}dG(v), r{rcub}, and such that for any two distinct colors i and j, every component of G[c --1({lcub}i,j{rcub})] must be a path. The linear list r-hued chromatic number of a graph G, denoted chiℓ L,r(G), is the smallest integer k such that for every k-list L, G has a linear (L,r)-coloring. Let Mad( G) denotes the maximum subgraph average degree of a graph G. We prove the following. (i) If G is a K3,3-minor free graph, then chi2(G) ≤ 5 and chi3(G) ≤ 10. Moreover, the bound of chi2( G) ≤ 5 is best possible. (ii) If G is a P4-free graph, then chir(G) ≤q chi( G) + 2(r -- 1), and this bound is best possible. (iii) If G is a P5-free bipartite graph, then chir( G) ≤ rchi(G), and this bound is best possible. (iv) If G is a P5-free graph, then chi2(G) ≤ 2chi(G), and this bound is best possible. (v) If G is a graph with maximum degree Delta, then each of the following holds. (i) If Delta ≥ 9 and Mad(G) \u3c 7/3, then chiℓL,r( G) ≤ max{lcub}lceil Delta/2 rceil + 1, r + 1{rcub}. (ii) If Delta ≥ 7 and Mad(G)\u3c 12/5, then chiℓ L,r(G)≤ max{lcub}lceil Delta/2 rceil + 2, r + 2{rcub}. (iii) If Delta ≥ 7 and Mad(G) \u3c 5/2, then chi ℓL,r(G)≤ max{lcub}lcei Delta/2 rceil + 3, r + 3{rcub}. (vi) If G is a K 4-minor free graph, then chiℓL,r( G) ≤ max{lcub}r,lceilDelta/2\rceil{rcub} + lceilDelta/2rceil + 2. (vii) Every planar graph G with maximum degree Delta has chiℓL,r(G) ≤ Delta + 7.;2. Results in digraph connectivity. For a graph G, let kappa( G), kappa\u27(G), delta(G) and tau( G) denote the connectivity, the edge-connectivity, the minimum degree and the number of edge-disjoint spanning trees of G, respectively. Let f(G) denote kappa(G), kappa\u27( G), or Delta(G), and define f¯( G) = max{lcub}f(H): H is a subgraph of G{rcub}. An edge cut X of a graph G is restricted if X does not contain all edges incident with a vertex in G. The restricted edge-connectivity of G, denoted by lambda2(G), is the minimum size of a restricted edge-cut of G. We define lambda 2(G) = max{lcub}lambda2(H): H ⊂ G{rcub}.;For a digraph D, let kappa;(D), lambda( D), delta--(D), and delta +(D) denote the strong connectivity, arc-strong connectivity, minimum in-degree, and out-degree of D, respectively. For each f ∈ {lcub}kappa,lambda, delta--, +{rcub}, define f¯(D) = max{lcub} f(H): H is a subdigraph of D{rcub}.;Catlin et al. in [Discrete Math., 309 (2009), 1033-1040] proved a characterization of kappa\u27(G) in terms of tau(G). We proved a digraph version of this characterization by showing that a digraph D is k-arc-strong if and only if for any vertex v in D, D has k-arc-disjoint spanning arborescences rooted at v. We also prove a characterization of uniformly dense digraphs analogous to the characterization of uniformly dense undirected graphs in [Discrete Applied Math., 40 (1992) 285--302]. (Abstract shortened by ProQuest.)

Cuts in matchings of 3-connected cubic graphs

We discuss conjectures on Hamiltonicity in cubic graphs (Tait, Barnette, Tutte), on the dichromatic number of planar oriented graphs (Neumann-Lara), and on even graphs in digraphs whose contraction is strongly connected (Hochst\"attler). We show that all of them fit into the same framework related to cuts in matchings. This allows us to find a counterexample to the conjecture of Hochst\"attler and show that the conjecture of Neumann-Lara holds for all planar graphs on at most 26 vertices. Finally, we state a new conjecture on bipartite cubic oriented graphs, that naturally arises in this setting.Comment: 12 pages, 5 figures, 1 table. Improved expositio

On Directed Feedback Vertex Set parameterized by treewidth

We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time $2^{o(t\log t)}\cdot n^{\mathcal{O}(1)}$ on general directed graphs, where $t$ is the treewidth of the underlying undirected graph. This is matched by a dynamic programming algorithm with running time $2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)}$. On the other hand, we show that if the input digraph is planar, then the running time can be improved to $2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}$.Comment: 20

Three-arc graphs: characterization and domination

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple $(v,u,x,y)$ of vertices such that both $(v,u,x)$ and $(u,x,y)$ are paths of length two. The 3-arc graph of a graph $G$ is defined to have vertices the arcs of $G$ such that two arcs $uv, xy$ are adjacent if and only if $(v,u,x,y)$ is a 3-arc of $G$. In this paper we give a characterization of 3-arc graphs and obtain sharp upper bounds on the domination number of the 3-arc graph of a graph $G$ in terms that of $G$