Let F be a family of graphs. A graph is F-free if it contains no copy of a graph in F as a subgraph. A cornerstone of extremal graph theory is the study of the Turán number ex(n,F), the maximum number of edges in an F-free graph on n vertices. Define the Zarankiewicz number z(n,F) to be the maximum number of edges in an F-free bipartite graph on n vertices. Let Ck denote a cycle of length k, an...

We study two extremal problems about subgraphs excluding a family F of graphs. i) Among all graphs with m edges, what is the smallest size f(m,F) of a largest F–free subgraph? ii) Among all graphs with minimum degree δ and maximum degree ∆, what is the smallest minimum degree h(δ,∆,F) of a spanning F– free subgraph with largest minimum degree? These questions are easy to answer for families not...

Interval orders and interval graphs are particularly natural examples of two widely studied classes of discrete structures: partially ordered sets and undirected graphs. So it is not surprising that researchers in such diverse fields as mathematics, computer science, engineering and the social sciences have investigated structural, algorithmic, enumerative, combinatorial, extremal and even expe...

We investigate the cop number of graphs based on combinatorial designs. Incidence graphs, point graphs, and block intersection graphs are studied, with an emphasis on finding families of graphs with large cop number. We generalize known results on so-called Meyniel extremal families by supplying bounds on the incidence graph of transversal designs, certain G-designs, and BIBDs with λ ≥ 1. Famil...

We study extremal problems about sets of integers that do not contain sumsets with summands of prescribed size. We analyse both finite sets and infinite sequences. We also study the connections of these problems with extremal problems of graphs and hypergraphs.

A multiple-interval representation of a simple graph G assigns each vertex a union of disjoint real intervals, such that vertices are adjacent if and only if their assigned sets intersect. The total interval number I (G) is the minimum of the total number of intervals used in any such representation of G. For triangle-free graphs, I (G) = m + t(G), where m is the number of edges in G and t(G) i...

This paper surveys extremal properties of general sum-connectivity index χα(G) in several classes of connected graphs of given order for some values of the parameter α: a) trees; b) connected unicyclic or bicyclic graphs; c) graphs of given connectivity. Work presented as invited lecture at CAIM 2014, September 19-22, “Vasile Alecsandri” University of Bacău, Romania.

In this paper, we construct new families of graphs whose automorphism groups are transitive on 3-paths. These graphs are constructed from certain Lie algebras related to generalized Kac-Moody algebras of rank two. We will show that one particular subfamily gives new lower bounds on the number of edges in extremal graphs with no cycles of length fourteen.

In this contribution, we first investigate sharp bounds for the reciprocal sum-degree distance of graphs with a given matching number. The corresponding extremal graphs are characterized completely. Then we explore the k-decomposition for the reciprocal sum-degree distance. Finally,we establish formulas for the reciprocal sum-degree distance of join and the Cartesian product of graphs.