TY - JOUR

T1 - Fractional Turan's theorem and bounds for the chromatic number

AU - Martínez, Leonardo

AU - Montejano, Luis

N1 - Funding Information:
1 Research supported by CONACyT-México under Project 166306 and under Scholarship 277462 2 Email: leomtz@im.unam.mx 3 Email: luismontej@gmail.com
Publisher Copyright:
© 2015 Elsevier B.V.

PY - 2015/12/1

Y1 - 2015/12/1

N2 - Turan's theorem gives conditions for finding large complete subgraphs in a graph in terms of the number of edges. In this work we look for families of graphs in which there is a similar theorem, but which will allow us to find larger complete subgraphs. This question is motivated by a geometric result by Katchalski and Liu concerning a fractional Helly theorem. We present results in two directions. On the one hand, we characterize the families of graphs for which there exists a constant β such that a proportion of α edges guarantees the existence of a complete subgraph using a proportion of αβ of the total number of vertices. On the other hand, we give a criterion to guarantee that in a family of graphs any fixed proportion of the edges is sufficient to conclude that the order of the largest complete subgraph goes to infinity as the number of vertices goes to infinity. These results are obtained by giving an upper bound for the chromatic number of a graph in terms of the clique number and an arbitrarily small proportion of the number of vertices.

AB - Turan's theorem gives conditions for finding large complete subgraphs in a graph in terms of the number of edges. In this work we look for families of graphs in which there is a similar theorem, but which will allow us to find larger complete subgraphs. This question is motivated by a geometric result by Katchalski and Liu concerning a fractional Helly theorem. We present results in two directions. On the one hand, we characterize the families of graphs for which there exists a constant β such that a proportion of α edges guarantees the existence of a complete subgraph using a proportion of αβ of the total number of vertices. On the other hand, we give a criterion to guarantee that in a family of graphs any fixed proportion of the edges is sufficient to conclude that the order of the largest complete subgraph goes to infinity as the number of vertices goes to infinity. These results are obtained by giving an upper bound for the chromatic number of a graph in terms of the clique number and an arbitrarily small proportion of the number of vertices.

KW - Chromatic graph theory

KW - Extremal graph theory

KW - Fractional Helly theorem

KW - Turan's theorem

UR - http://www.scopus.com/inward/record.url?scp=84953373889&partnerID=8YFLogxK

U2 - 10.1016/j.endm.2015.07.069

DO - 10.1016/j.endm.2015.07.069

M3 - Article

AN - SCOPUS:84953373889

VL - 50

SP - 415

EP - 420

JO - Electronic Notes in Discrete Mathematics

JF - Electronic Notes in Discrete Mathematics

SN - 1571-0653

ER -