The proof of a conjecture of Bouabdallah and Sotteau

Discussiones Mathematicae Graph Theory, 2005

The Directed Path Partition Conjecture is the following: If D is a digraph that contains no path with more than λ vertices then, for every pair (a, b) of positive integers with λ = a + b, there exists a vertex partition (A, B) of D such that no path in D A has more than a vertices and no path in D B has more than b vertices.We develop methods for finding the desired partitions for various classes of digraphs.

Canadian mathematical bulletin, 1968

Acta Mathematica Sinica, English Series, 2012

The induced path number ρ(G) of a graph G is defined as the minimum number of subsets into which the vertex set of G can be partitioned so that each subset induces a path. Broere et al. proved that if G is a graph of order n, then √ n ≤ ρ(G) + ρ(G) ≤ 3n 2. In this paper, we characterize the graphs G for which ρ(G) + ρ(G) = 3n 2 , improve the lower bound on ρ(G) + ρ(G) by one when n is the square of an odd integer, and determine a best possible upper bound for ρ(G) + ρ(G) when neither G nor G has isolated vertices.

The Electronic Journal of Combinatorics, 2005

The detour order of a graph G, denoted by (G) ; is the order of a longest path in G: A partition of the vertex set of G into two sets, A and B; such that (hAi) a and (hBi) b is called an (a; b)-partition of G.I fG has an (a; b)-partition for every pair (a; b) of positive

2006

The vertex set and arc set of a digraph D are denoted by V (D) and E (D), respectively, and the number of vertices in a digraph D is denoted by n (D). A directed cycle (path, walk) in a digraph will simply be called a cycle (path, walk). A graph or digraph is called hamiltonian if it ...

Networks, 1976

The n o t e , "On General Routing Problems" by J. K. Lenstra and A . H . G . Rinnooy Kan [ I ] , c o r r e c t s two e r r o r s i n r e c e n t papers by t h e author [2,31 on General Routing Problems, b u t i t tends t o obscure t h e r e a l i s s u e i n determining t h e complexity, o r d i f f i c u l t y , i n s o l v i n g a c t u a l r o u t i n g problems. The RPP, TSP, and GRP a r e a l l polynomial complete r o u t i n g problems, and t h a t i s p r e c i s e l y why t h e y r e q u i r e branch and bound ( s u b t o u r e l i m i n a t i o n ) a l g o r i t h m s , a s given i n [ 2 1 , which a r e n o t polynomial bounded. For such problems, an important approach i s t o reduce t h e complexity o f t h e problem a s much a s p o s s i b l e . The complexity of G R P problems depends n o t only on t h e number o f odd nodes and r e q u i r e d nodes (which may, f o r example, be t h e same f o r a TSP and CPP) b u t more i m p o r t a n t l y , on t h e number of disconnected components i n t h e problem ( t h e TSP has N while t h e CPP has only o n e ) . I t i s recommended i n [21 t h a t , a s f a r a s p o s s i b l e , r e q u i r e d nodes be converted t o r eq u i r e d a r c s because t h a t reduces t h e number of disconnected components i n t h e problem (because every r e q u i r e d node i s a disconnected component, while r e q u i r e d a r c s t e n d t o be conn e c t e d ) . The theorem p r e s e n t e d i n 111 , t h a t t h e RPP i s polynomial complete, i s t r u e b u t misleading. The RPP i s fundament a l l y more d i f f i c u l t than t h e CPP because it has disconnected components. The theorem i s obvious, s i n c e any TSP can be conv e r t e d t o an e q u i v a l e n t RPP by expanding any TSP node j t o a r e q u i r e d arc ( j , j') where j ' i s an a r t i f i c i a l d u p l i c a t e of j . Of course, t h i s t r a n s f o r m a t i o n from TSP t o RPP i s of no b e n e f i t , except a 2-matching problem on N nodes i s converted t o an e q u i v a l e n t 1-matching problem on 2 N nodes. On t h e o t h e r hand, an RPP w i t h only 2 disconnected components i s n o t much more d i f f i c u l t t o s o l v e than a CPP ( a t most one branch i n t h e branch and bound procedure i s r e q u i r e d ) . I n complexity o r d i f f i c u l t y , t h e RPP is somewhere between t h e CPP and t h e TSP, where t h e c r i t i c a l f a c t o r i s t h e number of disconnected components. Networks, 6: 281-284

Discrete Mathematics, 2002

For a graph G, let 2(G) denote the minimum degree sum of a pair of nonadjacent vertices. Suppose G is a graph of order n. Enomoto and Ota (J. Graph Theory 34 (2000) 163-169) conjectured that, if a partition n = k i=1 ai is given and 2(G) ¿ n + k − 1, then for any k distinct vertices v1; : : : ; v k , G can be decomposed into vertex-disjoint paths P1; : : : ; P k such that |V (Pi)| = ai and vi is an endvertex of Pi. Enomoto and Ota (J. Graph Theory 34 (2000) 163) veriÿed the conjecture for the case where all ai 6 5, and the case where k 6 3. In this paper, we prove the following theorem, with a stronger assumption of the conjecture. Suppose G is a graph of order n. If a partition n = k i=1 ai is given and 2(G) ¿ k i=1 max(4 3 ai ; ai + 1) − 1, then for any k distinct vertices v1; : : : ; v k , G can be decomposed into vertex-disjoint paths P1; : : : ; P k such that |V (Pi)| = ai and vi is an endvertex of Pi for all i. This theorem implies that the conjecture is true for the case where all ai 6 5 which was proved in (J. Graph Theory 34 (2000) 163-169).

2019

We look at some graph problems related to covering, partition, and connectivity. First, we study the problems of covering and partitioning edges with bicliques, especially from the viewpoint of parameterized complexity. For the partition problem, we develop much more efficient algorithms than the ones previously known. In contrast, for the cover problem, our lower bounds show that the known algorithms are probably optimal. Next, we move on to graph coloring, which is probably the most extensively studied partition problem in graphs. Hadwiger's conjecture is a long-standing open problem related to vertex coloring. We prove the conjecture for a special class of graphs, namely squares of 2-trees, and show that square graphs are important in connection with Hadwiger's conjecture. Then, we study a coloring problem that has been emerging recently, called rainbow coloring. This problem lies in the intersection of coloring and connectivity. We study different variants of rainbow coloring and present bounds and complexity results on them. Finally, we move on to another parameter related to connectivity called spanning tree congestion (STC). We give tight bounds for STC in general graphs and random graphs. While proving the results on STC, we also make some contributions to the related area of connected partitioning. Zusammenfassung Wir betrachten einige Graphprobleme mit Bezug auf Abdeckung, Partition und Konnektivität. Zunächst untersuchen wir Kantenabdeckung und-partition mit Bicliquen, insbesondere im Hinblick auf parametrisierte Komplexität. Für das Partitionierungsproblem entwickeln wir sehr viel effizientere Algorithmen als die bisher bekannten. Für das Abdeckungsproblem hingegen zeigen unsere unteren Schranken, dass die bereits bekannten Algorithmen wahrscheinlich optimal sind. Als Nächstes betrachten wir Graphfärbung, das wahrscheinlich am meisten untersuchte Partitionsproblem in Graphen. Hadwigers Vermutung ist ein seit Langem bestehendes offenes Problem bezüglich der Knotenfärbung. Wir beweisen diese Vermutung für eine spezielle Graphklasse, nämlich Quadrate von 2-Bäumen, und zeigen, dass Quadratgraphen wichtig in Bezug auf Hadwigers Vermutung sind. Danach untersuchen wir das Problem der sogenannten Regenbogenfärbung. Dieses erst kürzlich entstandene Problem liegt im Schnittpunkt der Probleme Färbung und Konnektivität. Wir untersuchen verschiedene Varianten der Regenbogenfärbung und zeigen Schranken und Komplexitätsergebnisse. Schließlich gehen wirüber zu einem weiteren Konnektivitätsparameter, der sogenannten spanning tree congestion (STC). Wir präsentieren scharfe Schranken für STC in allgemeinen und zufällig generierten Graphen. Unsere Ergebnisse in diesem Bereich leisten darüberhinaus einen Beitrag zu dem verwandten Gebiet der verbundenen Partitionierung. vi First of all I would like to thank my advisor Andreas Karrenbauer. I thank him for accepting me as a PhD student and guiding me through my PhD for the past 4 years. He has always given me the freedom to follow my own research interests, at the same time making sure that I don't go much out of the track. I thank him for his contributions, technical and otherwise to my thesis. I was very lucky to have Prof. Sunil Chandran visiting MPI-Informatics during my PhD term for about 18 months. He has been a co-author in most of my publications and has had a big effect on my shaping my research outlook. It has been a great pleasure to collaborate with him. I thank Erik Jan for all the technical input and for the encouragement he has given me. I thank Prof. Ragesh Jaiswal, who was my Masters supervisor for being a mentor at the start of my research career. I thank all my other co-authors Anita, Marco, Juho, Paloma, Pinar, and Sanming for their contributions. I thank Kurt Mehlhorn, for giving me the opportunity to be a PhD student at the Algorithms department, and also for building such an encouraging atmosphere in the group. I thank Pavel, for being the most friendly and helpful officemate for 4 years. I think I have learned a lot from Pavel. I also thank the other current and past members of D1:

2012

Journal of Mathematics, 2019

Simple finite connected graphs G=V,E of p≥2 vertices are considered in this paper. A connected detour set of G is defined as a subset S⊆V such that the induced subgraph GS is connected and every vertex of G lies on a u−v detour for some u,v∈S. The connected detour number cdnG of a graph G is the minimum order of the connected detour sets of G. In this paper, we determined cdnG for three special classes of graphs G, namely, unicyclic graphs, bicyclic graphs, and cog-graphs for Cp, Kp, and Km,n.