Florian Pfender

Vertex-coloring edge-weightings: Towards the 1-2-3-conjecture

A weighting of the edges of a graph is called vertex-coloring if the weighted degrees of the vertices yield a proper coloring of the graph. In this paper we show that such a weighting is possible from the weight set {1,2,3,4,5} for all graphs not containing components with exactly 2 vertices.

A New Upper Bound for the Irregularity Strength of Graphs

A weighting of the edges of a graph is called irregular if the weighted degrees of the vertices are all different. In this note we show that such a weighting is possible from the weight set {1,2,…,...

Maximum density of induced 5-cycle is achieved by an iterated blow-up of 5-cycle

Let C ( n ) denote the maximum number of induced copies of 5 -cycles in graphs on n vertices. For n large enough, we show that C ( n ) = a ? b ? c ? d ? e + C ( a ) + C ( b ) + C ( c ) + C ( d ) + C ( e ) , where a + b + c + d + e = n and a , b , c , d , e are as equal as possible.Moreover, for n a power of 5, we show that the unique graph on n vertices maximizing the number of induced 5 -cycles is an iterated blow-up of a 5-cycle.The proof uses flag algebra computations and stability methods.

Claw-free 3-connected P11-free graphs are hamiltonian

We show that every 3-connected claw-free graph which contains no induced copy of P11 is hamiltonian. Since there exist non-hamiltonian 3-connected claw-free graphs without induced copies of P12 this result is, in a way, best possible.

Linear forests and ordered cycles

A collection L = P 1 ∪ P 2 ∪ · · · ∪ P t (1 ≤ t ≤ k) of t disjoint paths, s of them being singletons with |V (L)| = k is called a (k, t, s)-linear forest. A graph G is (k, t, s)ordered if for every (k, t, s)-linear forest L in G there exists a cycle C in G that contains the paths of L in the designated order as subpaths. If the cycle is also a hamiltonian cycle, then G is said to be (k, t, s)-ordered hamiltonian. We give sharp sum of degree conditions for nonadjacent vertices that imply a graph is (k, t, s)-ordered hamiltonian.

Extremal graphs for intersecting cliques

For any two positive integers n?r?1, the well-known Turan Theorem states that there exists a least positive integer ex(n,Kr) such that every graph with n vertices and ex(n,Kr)+1 edges contains a subgraph isomorphic to Kr. We determine the minimum number of edges sufficient for the existence of k cliques with r vertices each intersecting in exactly one common vertex.

Improved Delsarte bounds for spherical codes in small dimensions

We present an extension of the Delsarte linear programming method for spherical codes. For several dimensions it yields improved upper bounds including some new bounds on kissing numbers. Musins recent work on kissing numbers in dimensions three and four can be formulated in our framework.

New Ore-Type Conditions for H -Linked Graphs

For a fixed (multi)graph H, a graph G is H-linked if any injection f: V(H)→V(G) can be extended to an H-subdivision in G. The notion of an H -linked graph encompasses several familiar graph classes, including k-linked, k-ordered and k-connected graphs. In this article, we give two sharp Ore-type degree sum conditions that assure a graph G is H -linked for arbitrary H. These results extend and refine several previous results on H -linked, k-linked, and k-ordered graphs.

Complexity results for rainbow matchings

A rainbow matching in an edge-colored graph is a matching whose edges have distinct colors. We address the complexity issue of the following problem, max rainbow matching: Given an edge-colored graph G, how large is the largest rainbow matching in G? We present several sharp contrasts in the complexity of this problem.We show, among others, thatmax rainbow matching can be approximated by a polynomial algorithm with approximation ratio 2 / 3 - e .max rainbow matching is APX-complete, even when restricted to properly edge-colored linear forests without a 5-vertex path, and is solvable in polynomial time for edge-colored forests without a 4-vertex path.max rainbow matching is APX-complete, even when restricted to properly edge-colored trees without an 8-vertex path, and is solvable in polynomial time for edge-colored trees without a 7-vertex path.max rainbow matching is APX-complete, even when restricted to properly edge-colored paths. These results provide a dichotomy theorem for the complexity of the problem on forests and trees in terms of forbidding paths. The latter is somewhat surprising, since, to the best of our knowledge, no (unweighted) graph problem prior to our result is known to be NP-hard for simple paths.We also address the parameterized complexity of the problem.

Cycle spectra of Hamiltonian graphs

We prove that every Hamiltonian graph with n vertices and m edges has cycles with more than p-12lnp-1 different lengths, where p=m-n. For general m and n, there exist such graphs having at most 2@?p+1@? different cycle lengths.