Some Results on incidence coloring, star arboricity and domination number

Two inequalities bridging the three isolated graph invariants, incidence chromatic number, star arboricity and domination number, were established. Consequently, we deduced an upper bound and a lower bound of the incidence chromatic number for all graphs. Using these bounds, we further reduced the upper bound of the incidence chromatic number of planar graphs and showed that cubic graphs with orders not divisible by four are not 4-incidence colorable. The incidence chromatic numbers of Cartesian product, join and union of graphs were also determined.Comment: 8 page

Exploiting c\mathbf{c}-Closure in Kernelization Algorithms for Graph Problems

A graph is c-closed if every pair of vertices with at least c common neighbors is adjacent. The c-closure of a graph G is the smallest number such that G is c-closed. Fox et al. [ICALP '18] defined c-closure and investigated it in the context of clique enumeration. We show that c-closure can be applied in kernelization algorithms for several classic graph problems. We show that Dominating Set admits a kernel of size k^O(c), that Induced Matching admits a kernel with O(c^7*k^8) vertices, and that Irredundant Set admits a kernel with O(c^(5/2)*k^3) vertices. Our kernelization exploits the fact that c-closed graphs have polynomially-bounded Ramsey numbers, as we show

Problems and memories

I state some open problems coming from joint work with Paul Erd\H{o}sComment: This is a paper form of the talk I gave on July 5, 2013 at the centennial conference in Budapest to honor Paul Erd\H{o}

Solving Hard Graph Problems with Combinatorial Computing and Optimization

Many problems arising in graph theory are difficult by nature, and finding solutions to large or complex instances of them often require the use of computers. As some such problems are $NP$-hard or lie even higher in the polynomial hierarchy, it is unlikely that efficient, exact algorithms will solve them. Therefore, alternative computational methods are used. Combinatorial computing is a branch of mathematics and computer science concerned with these methods, where algorithms are developed to generate and search through combinatorial structures in order to determine certain properties of them. In this thesis, we explore a number of such techniques, in the hopes of solving specific problem instances of interest. Three separate problems are considered, each of which is attacked with different methods of combinatorial computing and optimization. The first, originally proposed by ErdH{o}s and Hajnal in 1967, asks to find the Folkman number $F_e(3,3;4)$, defined as the smallest order of a $K_4$-free graph that is not the union of two triangle-free graphs. A notoriously difficult problem associated with Ramsey theory, the best known bounds on it prior to this work were $19 leq F_e(3,3;4) leq 941$. We improve the upper bound to $F_e(3,3;4) leq 786$ using a combination of known methods and the Goemans-Williamson semi-definite programming relaxation of MAX-CUT. The second problem of interest is the Ramsey number $R(C_4,K_m)$, which is the smallest $n$ such that any $n$-vertex graph contains a cycle of length four or an independent set of order $m$. With the help of combinatorial algorithms, we determine $R(C_4,K_9)=30$ and $R(C_4,K_{10})=36$ using large-scale computations on the Open Science Grid. Finally, we explore applications of the well-known Lenstra-Lenstra-Lov\u27{a}sz (LLL) algorithm, a polynomial-time algorithm that, when given a basis of a lattice, returns a basis for the same lattice with relatively short vectors. The main result of this work is an application to graph domination, where certain hard instances are solved using this algorithm as a heuristic