Meike Tewes

Three-colourbility and forbidden subgraphs. II: polynomial algorithms

In this paper we study the chromatic number for graphs with forbidden induced subgraphs. We focus our interest on graph classes (defined in terms of forbidden induced subgraphs) for which the question of 3-colourability can be decided in polynomial time and, if so, a proper 3-colouring can be determined also in polynomial time. Note that the 3-colourability decision problem is a well-known NP-complete problem, even for special graph classes, e.g. triangle-free and K1,5-free (Discrete Math. 162 (1-3) (1996) 313). Therefore, it is unlikely that there exists a polynomial algorithm deciding whether there exists a 3-colouring of a given graph in general. We present three different approaches to reach our goal. The first approach is purely a structural analysis of the graph class in consideration; the second one is a structural analysis of only the non-perfect K4-free members of the considered graph class; finally the last approach is based on propositional logic and bounded dominating subgraphs.

Vertex pancyclic graphs

Let G be a graph of order n. A graph G is called pancyclic if it contains a cycle of length k for every 3 ≤ k ≤ n, and it is called vertex pancyclic if every vertex is contained in a cycle of length k for every 3 ≤ k ≤ n. In this paper, we shall present different sufficient conditions for graphs to be vertex pancyclic.

Almost all almost regular c -partite tournaments with c ≥ 5 are vertex pancyclic

Abstract A tournament is an orientation of a complete graph and a multipartite or c-partite tournament is an orientation of a complete c-partite graph. If D is a digraph, then let d + ( x ) be the outdegree and d − ( x ) the indegree of the vertex x in D. The minimum (maximum) outdegree and the minimum (maximum) indegree of D are denoted by δ + ( Δ + ) and δ − ( Δ − ), respectively. In addition, we define δ =min{ δ + , δ − } and Δ =max{ Δ + , Δ − }. A digraph is regular when δ = Δ and almost regular when Δ − δ ⩽1. Recently, the third author proved that all regular c-partite tournaments are vertex pancyclic when c ⩾5, and that all, except possibly a finite number, regular 4-partite tournaments are vertex pancyclic. Clearly, in a regular multipartite tournament, each partite set has the same cardinality. As a supplement of Yeos result we prove first that an almost regular c-partite tournament with c ⩾5 is vertex pancyclic, if all partite sets have the same cardinality. Second, we show that all almost regular c-partite tournaments are vertex pancyclic when c ⩾8, and third that all, except possibly a finite number, almost regular c-partite tournaments are vertex pancyclic when c ⩾5.

Vertex deletion and cycles in multipartite tournaments

Abstract A tournament is an orientation of a complete graph and a multipartite tournament is an orientation of a complete multipartite graph. Therefore, a tournament is a k -partite tournament with exactly k -vertices. From the well-known theorem of Moon that every vertex of a strong tournament T is contained in a directed cycle of length m for 3 ⩽ m ⩽| V ( T ), it follows easily that T has at least two vertices u 1 and u 2 such that T − u i is strong for i = 1, 2, if | V ( T )| ⩾ 4. As a generalization of this statement, we prove in this paper that all strongly connected multipartite tournaments D of order | V ( D )|⩾ 4 have two different vertices u 1 and u 2 such that D − u 1 is strong for i = 1, 2, with exception of a well determined family of bipartite tournaments and three well determined families of 3-partite tournaments. In addition, we show that the special class of Hamiltonian k -partite tournaments D of order n ⩾ 5 that are not 2-connected, contains a directed cycle C of length p or p − 1 for every 4 ⩽ p ⩽ n such that the induced subdigraph D [ V ( C )] is not 2-connected. Furthermore, every vertex of such a digraph is contained in a cycle of length n − 1 or n − 2.

Vertex pancyclic in-tournaments

An in-tournament is an oriented graph such that the negative neighborhood of every vertex induces a tournament. The topic of this paper is to investigate vertex k-pancyclicity of in-tournaments of order n, where for some 3 ≤ k ≤ n, every vertex belongs to a cycle of length p for every k ≤ p ≤ n. We give sharp lower bounds for the minimum degree such that a strong in-tournament is vertex k-pancyclic for k ≤ 5 and k ≥ n - 3. In the latter case, we even show that the in-tournaments in consideration are fully (n - 3)-extendable which means that every vertex belongs to a cycle of length n - 3 and that the vertex set of every cycle of length at least n - 3 is contained in a cycle of length one greater. In accordance with these results, we state the conjecture that every strong in-tournament of order n with minimum degree greater than

Pancyclic in-tournaments

{{9(n-k-1)}\over{5+6k+(-1)^k2^{-k+2}}}+1

Longest Paths and Longest Cycles in Graphs with Large Degree Sums

is vertex k-pancyclic for 5 < k < n - 3, and we present a family of examples showing that this bound would be best possible.

Longest paths in strong spanning oriented subgraphs of strong semicomplete multipartite digraphs

Abstract An in-tournament is an oriented graph, where the negative neighborhood of every vertex induces a tournament. In this paper, the influence of the minimum indegree δ − ( D ) of an in-tournament D on its k-pancyclicity is considered. An oriented graph of order n is said to be k-pancyclic for some 3⩽ k ⩽ n , if it contains an oriented cycle of length t for every k ⩽ t ⩽ n . For every 3⩽ k ⩽ n , a lower bound for δ − ( D ) is presented that ensures a strong in-tournament to be k-pancyclic. Examples show that all bounds given here are best possible.

The ratio of the longest cycle and longest path on semicomplete mutlipartitite digraphs

Abstract. Let p(G) and c(G) denote the number of vertices in a longest path and a longest cycle, respectively, of a finite, simple graph G. Define σ4(G)=min{d(x1)+d(x2)+ d(x3)+d(x4) | {x1,…,x4} is independent in G}. In this paper, the difference p(G)−c(G) is considered for 2-connected graphs G with σ4(G)≥|V(G)|+3. Among others, we show that p(G)−c(G)≤2 or every longest path in G is a dominating path.

Vertex Pancyclic Graphs

Abstract A digraph obtained by replacing each edge of a complete multipartite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete multipartite digraph. Volkmann (Manuscript, RWTH Aachen, Germany, June 1998) raised the following question: Let D be a strong semicomplete multipartite digraph with a longest path of length l. Does there exist a strong spanning oriented subgraph of D with a longest path of length l? We provide examples which show that the answer to this question is negative. We also demonstrate that every strong semicomplete multipartite digraph D, which is not bipartite with a partite set of cardinality one, has a strong spanning oriented subgraph of D with a longest path of length at least l −2. This bound is sharp.