58 research outputs found
GráfszĂnezĂ©sek Ă©s gráfok felbontásai = Colorings and decompositions of graphs
A nem-ismĂ©tlĹ‘ szĂnezĂ©seket a vĂ©letlen mĂłdszer alkalmazhatĂłsága miatt kezdtĂ©k el vizsgálni. FelsĹ‘ korlátot adtunk a szĂnek számára, amely a maximum fok Ă©s a favastagság lineáris fĂĽggvĂ©nye. Olyan szĂnezĂ©seket is vizsgáltunk, amelyek egy sĂkgráf oldalain nem-ismĂ©tlĹ‘k. SejtĂ©s volt, hogy vĂ©ges sok szĂn elĂ©g. Ezt bizonyĂtottuk 24 szĂnnel. A kromatikus számot Ă©s a metszĂ©si számot algoritmikusan nehĂ©z meghatározni. EzĂ©rt meglepĹ‘ Albertson egy friss sejtĂ©se, amely kapcsolatot állĂt fel közöttĂĽk: ha egy gráf kromatikus száma r, akkor metszĂ©si száma legalább annyi, mint a teljes r csĂşcsĂş gráfĂ©. BizonyĂtottuk a sejtĂ©st, ha r<3.57n, valamint ha 12<r<17. Ez utĂłbbi azĂ©rt Ă©rdekes, mert a teljes r csĂşcsĂş gráf metszĂ©si száma csak r<13 esetĂ©n ismert. A tĂ©makör legfontosabb nyitott kĂ©rdĂ©se a Hadwiger-sejtĂ©s, mely szerint minden r-kromatikus gráf tartalmazza a teljes r csĂşcsĂş gráfot minorkĂ©nt. Ennek általánosĂtásakĂ©nt fogalmazták meg a lista szĂnezĂ©si Hadwiger sejtĂ©st: ha egy gráf nem tartalmaz teljes r csĂşcsĂş gráfot minorkĂ©nt, akkor az r-lista szĂnezhetĹ‘. Megmutattuk, hogy ez a sejtĂ©s hamis. Legalább cr szĂnre szĂĽksĂ©gĂĽnk van bizonyos gráfokra, ahol c=4/3. Thomassennel vetettĂĽk fel azt a kĂ©rdĂ©st, hogy milyen feltĂ©tel garantálja, hogy G Ă©lei felbonthatĂłk egy adott T fa pĂ©ldányaira. Legyen Y az a fa, melynek fokszámsorozata (1,1,1,2,3). Megmutattuk, hogy minden 287-szeresen Ă©lösszefĂĽggĹ‘ fa felbomlik Y-okra, ha az Ă©lszám oszthatĂł 4-gyel. | Nonrepetitive colorings often use the probabilistic method. We gave an upper bound as a linear function of the maximum degree and the tree-width. We also investigated colorings, which are nonrepetitive on faces of plane graphs. As conjectured, a finite number of colors suffice. We proved it by 24 colors. The chromatic and crossing numbers are both difficult to compute. The recent Albertson's conjecture is a surprising relation between the two: if the chromatic number is r, then the crossing number is at least the crossing number of the complete graph on r vertices. We proved this claim, if r<3.57n, or 12<r<17. The latter is remarkable, since the crossing number of the complete graph is only known for r<13. The most important open question of the field is Hadwiger's conjecture: every r-chromatic graph contains as a minor the complete graph on r vertices. As a generalisation, the following is the list coloring Hadwiger conjecture: if a graph does not contain as a minor the complete graph on r vertices , then the graph is r-list colorable. We proved the falsity of this claim. In our examples, at least cr colors are necessary, where c=4/3. Decomposition of graphs is well-studied. Thomassen and I posed the question of a sufficient connectivity condition, which guaranties a T-decomposition. Let Y be the tree with degree sequence (1,1,1,2,3). We proved every 287-edge connected graph has a Y-decomposition, if the size is divisible by four
Forbidden Configurations: Finding the number predicted by the Anstee-Sali Conjecture is NP-hard
Let F be a hypergraph and let forb(m,F) denote the maximum number of edges a
hypergraph with m vertices can have if it doesn't contain F as a subhypergraph.
A conjecture of Anstee and Sali predicts the asymptotic behaviour of forb(m,F)
for fixed F. In this paper we prove that even finding this predicted asymptotic
behaviour is an NP-hard problem, meaning that if the Anstee-Sali conjecture
were true, finding the asymptotics of forb(m,F) would be NP-hard
The smallest 5-chromatic tournament
A coloring of a digraph is a partition of its vertex set such that each class
induces a digraph with no directed cycles. A digraph is -chromatic if is
the minimum number of classes in such partition, and a digraph is oriented if
there is at most one arc between each pair of vertices. Clearly, the smallest
-chromatic digraph is the complete digraph on vertices, but determining
the order of the smallest -chromatic oriented graphs is a challenging
problem. It is known that the smallest -, - and -chromatic oriented
graphs have , and vertices, respectively. In 1994, Neumann-Lara
conjectured that a smallest -chromatic oriented graph has vertices. We
solve this conjecture and show that the correct order is
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