Papers by Florian Pfender
Electronic Journal of Combinatorics, Jun 28, 2012
A rainbow matching in an edge-colored graph is a matching in which all the edges have distinct co... more A rainbow matching in an edge-colored graph is a matching in which all the edges have distinct colors. Wang asked if there is a function f (δ) such that a properly edgecolored graph G with minimum degree δ and order at least f (δ) must have a rainbow matching of size δ. We answer this question in the affirmative; an extremal approach yields that f (δ) = 98δ/23 < 4.27δ suffices. Furthermore, we give an O(δ(G)|V (G)| 2)time algorithm that generates such a matching in a properly edge-colored graph of order at least 6.5δ.
arXiv (Cornell University), Apr 14, 2012
A rainbow subgraph of an edge-colored graph is a subgraph whose edges have distinct colors. The c... more A rainbow subgraph of an edge-colored graph is a subgraph whose edges have distinct colors. The color degree of a vertex v is the number of different colors on edges incident to v. We show that if n is large enough (namely, n ≥ 4.25k 2), then each n-vertex graph G with minimum color degree at least k contains a rainbow matching of size at least k.
Journal of Graph Theory, Nov 1, 2009
Let F k be the family of graphs G such that all sufficiently large k-connected claw-free graphs w... more Let F k be the family of graphs G such that all sufficiently large k-connected claw-free graphs which contain no induced copies of G are subpancyclic. We show that for every k ≥ 3 the family F k is infinite and make the first step towards the complete characterization of the family F 3 .
Journal of Graph Theory, 2004
We characterize all pairs of connected graphs {X, Y } such that each 3-connected {X, Y }-free gra... more We characterize all pairs of connected graphs {X, Y } such that each 3-connected {X, Y }-free graph is pancyclic. In particular, we show that if each of the graphs in such a pair {X, Y } has at least four vertices, then one of them is the claw K 1,3 , while the other is a subgraph of one of six specified graphs.
arXiv (Cornell University), Nov 3, 2015
Motivated by investigations of rainbow matchings in edge colored graphs, we introduce the notion ... more Motivated by investigations of rainbow matchings in edge colored graphs, we introduce the notion of color-line graphs that generalizes the classical concept of line graphs in a natural way. Let H be a (properly) edge colored graph. The (proper) color-line graph CL(H) of H has edges of H as vertices, and two edges of H are adjacent in CL(H) if they have an endvertex in common or have the same color. We give Krausz-type characterizations for (proper) color-line graphs, and show that, for any fixed k, recognizing color-line graphs of properly edge colored graphs H with at most k colors is polynomially solvable. Moreover, we give a good characterization for proper 2-color-line graphs that yields a linear time recognition algorithm in this case. In contrast, we point out that, for any fixed k ≥ 2, recognizing if a graph is the color-line graph of some graph H in which the edges are colored with at most k colors is NP-complete.
SIAM Journal on Discrete Mathematics, 2015
For a given multigraph H, a graph G is H-linked, if |G| ≥ |H| and for every injective map τ : V (... more For a given multigraph H, a graph G is H-linked, if |G| ≥ |H| and for every injective map τ : V (H) → V (G), we can find internally disjoint paths in G, such that every edge from uv in H corresponds to a τ (u) − τ (v) path. To guarantee that a G is H-linked, you need a minimum degree larger than |G| 2. This situation changes, if you know that G has a certain connectivity k. Depending on k, even a minimum degree independent of |G| may suffice. Let δ(k, H, N) be the minimum number, such that every k-connected graph G with |G| = N and δ(G) ≥ δ(k, H, N) is H-linked. We study bounds for this quantity. In particular, we find bounds for all multigraphs H with at most three edges, which are optimal up to small additive or multiplicative constants. Fact 1.1. Let H 1 and H 2 be multigraphs and suppose that H 2 is a submultigraph of H 1. Then every H 1-linked graph is H 2-linked. Fact 1.2. Let H 1 and H 2 be multigraphs and suppose that one gets H 2 from H 1 through the identification of two non-adjacent vertices, one of which has degree 1. Then every H 1-linked graph is H 2-linked. Corollary 1.3. Let H be a multigraph without isolated vertices. Then every |E(H)|-linked graph is H-linked. Fact 1.4. Let H be a multigraph with a k-multi-edge. Then, every H-linked graph is (|H| − 2 + k)connected.
arXiv (Cornell University), Apr 1, 2018
arXiv (Cornell University), Apr 16, 2018
In 1935, Erdős and Szekeres proved that (m − 1)(k − 1) + 1 is the minimum number of points in the... more In 1935, Erdős and Szekeres proved that (m − 1)(k − 1) + 1 is the minimum number of points in the plane which definitely contain an increasing subset of m points or a decreasing subset of k points (as ordered by their x-coordinates). We consider their result from an on-line game perspective: Let points be determined one by one by player A first determining the x-coordinate and then player B determining the y-coordinate. What is the minimum number of points such that player A can force an increasing subset of m points or a decreasing subset of k points? We introduce this as the Erdős-Szekeres on-line number and denote it by ESO(m, k). We observe that ESO(m, k) < (m − 1)(k − 1) + 1 for m, k ≥ 3, provide a general lower bound for ESO(m, k), and determine ESO(m, 3) up to an additive constant.
arXiv (Cornell University), Sep 30, 2019
The Erdős-Simonovits stability theorem states that for all ε > 0 there exists α > 0 such that if ... more The Erdős-Simonovits stability theorem states that for all ε > 0 there exists α > 0 such that if G is a K r+1-free graph on n vertices with e(G) > ex(n, K r+1) − αn 2 , then one can remove εn 2 edges from G to obtain an r-partite graph. Füredi gave a short proof that one can choose α = ε. We give a bound for the relationship of α and ε which is asymptotically sharp as ε → 0.
arXiv (Cornell University), Mar 10, 2011
We investigate a graph parameter called the total vertex irregularity strength (tvs(G)), i.e. the... more We investigate a graph parameter called the total vertex irregularity strength (tvs(G)), i.e. the minimal s such that there is a labeling w : E(G) ∪ V (G) → {1, 2,. .. , s} of the edges and vertices of G giving distinct weighted degrees wt G (v) := w(v) + v∈e∈E(G) w(e) for every pair of vertices of G. We prove that tvs(F) = ⌈(n 1 + 1)/2⌉ for every forest F with no vertices of degree 2 and no isolated vertices, where n 1 is the number of pendant vertices in F. Stronger results for trees were recently proved by Nurdin et al.
Journal of Graph Theory, Jul 6, 2021
We study edge-maximal, non-complete graphs on surfaces that do not triangulate the surface. We pr... more We study edge-maximal, non-complete graphs on surfaces that do not triangulate the surface. We prove that there is no such graph on the projective plane N 1 , K 7 − e is the unique such graph on the Klein bottle N 2 and K 8 − E(C 5) is the unique such graph on the torus S 1. In contrast to this for each g ≥ 2 we construct an infinite family of such graphs on the orientable surface S g of genus g, that are ⌊ g 2 ⌋ edges short of a triangulation.
arXiv (Cornell University), Feb 10, 2015
An edge-ordering of a graph G = (V, E) is a bijection φ : E → {1, 2,. .. , |E|}. Given an edge-or... more An edge-ordering of a graph G = (V, E) is a bijection φ : E → {1, 2,. .. , |E|}. Given an edge-ordering, a sequence of edges P = e 1 , e 2 ,. .. , e k is an increasing path if it is a path in G which satisfies φ(e i) < φ(e j) for all i < j. For a graph G, let f (G) be the largest integer ℓ such that every edge-ordering of G contains an increasing path of length ℓ. The parameter f (G) was first studied for G = K n and has subsequently been studied for other families of graphs. This paper gives bounds on f for the hypercube and the random graph G(n, p).
arXiv (Cornell University), Oct 14, 2016
Bootstrap percolation is a deterministic cellular automaton in which vertices of a graph G begin ... more Bootstrap percolation is a deterministic cellular automaton in which vertices of a graph G begin in one of two states, "dormant" or "active". Given a fixed positive integer r, a dormant vertex becomes active if at any stage it has at least r active neighbors, and it remains active for the duration of the process. Given an initial set of active vertices A, we say that G rpercolates (from A) if every vertex in G becomes active after some number of steps. Let m(G, r) denote the minimum size of a set A such that G r-percolates from A. Bootstrap percolation has been studied in a number of settings, and has applications to both statistical physics and discrete epidemiology. Here, we are concerned with degree-based density conditions that ensure m(G, 2) = 2. In particular, we give an Ore-type degree sum result that states that if a graph G satisfies σ 2 (G) ≥ n − 2, then either m(G, 2) = 2 or G is in one of a small number of classes of exceptional graphs. (Here, σ 2 (G) is the minimum sum of degrees of two non-adjacent vertices in G.) We also give a Chvátal-type degree condition: If G is a graph with degree sequence d 1 ≤ d 2 ≤ • • • ≤ d n such that d i ≥ i + 1 or d n−i ≥ n − i − 1 for all 1 ≤ i < n 2 , then m(G, 2) = 2 or G falls into one of several specific exceptional classes of graphs. Both of these results are inspired by, and extend, an Ore-type result in [D. Freund, M. Poloczek, and D. Reichman, Contagious sets in dense graphs, European J. Combin. 68 2018].
Journal of Graph Theory, Mar 22, 2016
The crossing number cr(G) of a graph G is the minimum number of crossings in a nondegenerate plan... more The crossing number cr(G) of a graph G is the minimum number of crossings in a nondegenerate planar drawing of G. The rectilinear crossing number cr(G) of G is the minimum number of crossings in a rectilinear nondegenerate planar drawing (with edges as straight line segments) of G. Zarankiewicz proved in 1952 that cr(K n 1
European Journal of Combinatorics, Dec 1, 2018
For all n ≥ 9, we show that the only triangle-free graphs on n vertices maximizing the number 5-c... more For all n ≥ 9, we show that the only triangle-free graphs on n vertices maximizing the number 5-cycles are balanced blow-ups of a 5-cycle. This completely resolves a conjecture by Erdős, and extends results by Grzesik and Hatami, Hladký, Král', Norin, and Razborov, where they independently showed this same result for large n and for all n divisible by 5.
Journal of Combinatorial Theory, Series B, Sep 1, 2017
Erdős and Sós proposed a problem of determining the maximum number F (n) of rainbow triangles in ... more Erdős and Sós proposed a problem of determining the maximum number F (n) of rainbow triangles in 3-edge-colored complete graphs on n vertices. They conjectured that F (n) = F (a)+ F (b) + F (c) + F (d) + abc + abd + acd + bcd, where a + b + c + d = n and a, b, c, d are as equal as possible. We prove that the conjectured recurrence holds for sufficiently large n. We also prove the conjecture for n = 4 k for all k ≥ 0. These results imply that lim F (n) (n 3) = 0.4, and determine the unique limit object. In the proof we use flag algebras combined with stability arguments.
Discrete and Computational Geometry, Mar 1, 2008
The visibility graph V(X) of a discrete point set X ⊂ R 2 has vertex set X and an edge xy for eve... more The visibility graph V(X) of a discrete point set X ⊂ R 2 has vertex set X and an edge xy for every two points x, y ∈ X whenever there is no other point in X on the line segment between x and y. We show that for every graph G, there is a point set X ∈ R 2 , such that the subgraph of V(X ∪ Z 2) induced by X is isomorphic to G. As a consequence, we show that there are visibility graphs of arbitrary high chromatic number with clique number 6 settling a question by Kára, Pór and Wood.
arXiv (Cornell University), Dec 21, 2013
A total weighting of the vertices and edges of a hypergraph is called vertex-coloring if the tota... more A total weighting of the vertices and edges of a hypergraph is called vertex-coloring if the total weights of the vertices yield a proper coloring of the graph, i.e., every edge contains at least two vertices with different weighted degrees. In this note we show that such a weighting is possible if every vertex has two, and every edge has three weights to choose from, extending a recent result on graphs to hypergraphs.
arXiv (Cornell University), Jun 23, 2010
Let m ∶= E(G) sufficiently large and s ∶= ⌈ m−1 3 ⌉. We show that unless the maximum degree ∆ > 2... more Let m ∶= E(G) sufficiently large and s ∶= ⌈ m−1 3 ⌉. We show that unless the maximum degree ∆ > 2s, there is a weightingŵ ∶ E ∪ V → {0, 1,. .. , s} so thatŵ(uv) +ŵ(u) +ŵ(v) ≠ w(u ′ v ′) +ŵ(u ′) +ŵ(v ′) whenever uv ≠ u ′ v ′ (such a weighting is called total edge irregular). This validates a conjecture by Ivančo and Jendrol' for large graphs, extending a result by Brandt, Miškuf and Rautenbach.
arXiv (Cornell University), Aug 13, 2014
Let G be a fixed graph and let F be a family of graphs. A subgraph J of G is F-saturated if no me... more Let G be a fixed graph and let F be a family of graphs. A subgraph J of G is F-saturated if no member of F is a subgraph of J, but for any edge e in E(G) − E(J), some element of F is a subgraph of J + e. We let ex(F, G) and sat(F, G) denote the maximum and minimum size of an F-saturated subgraph of G, respectively. If no element of F is a subgraph of G, then sat(F, G) = ex(F, G) = |E(G)|. In this paper, for k ≥ 3 and n ≥ 100 we determine sat(K 3 , K n k), where K n k is the complete balanced k-partite graph with partite sets of size n. We also give several families of constructions of K t-saturated subgraphs of K n k for t ≥ 4. Our results and constructions provide an informative contrast to recent results on the edge-density version of ex(K t , K n k) from [A.
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Papers by Florian Pfender