Statistical Mechanics of maximal independent sets

The graph theoretic concept of maximal independent set arises in several practical problems in computer science as well as in game theory. A maximal independent set is defined by the set of occupied nodes that satisfy some packing and covering constraints. It is known that finding minimum and maximum-density maximal independent sets are hard optimization problems. In this paper, we use cavity method of statistical physics and Monte Carlo simulations to study the corresponding constraint satisfaction problem on random graphs. We obtain the entropy of maximal independent sets within the replica symmetric and one-step replica symmetry breaking frameworks, shedding light on the metric structure of the landscape of solutions and suggesting a class of possible algorithms. This is of particular relevance for the application to the study of strategic interactions in social and economic networks, where maximal independent sets correspond to pure Nash equilibria of a graphical game of public goods allocation

On Constrained Intersection Representations of Graphs and Digraphs

We study the problem of determining minimal directed intersection representations of DAGs in a model introduced by [Kostochka, Liu, Machado, and Milenkovic, ISIT2019]: vertices are assigned color sets, two vertices are connected by an arc if and only if they share at least one color and the tail vertex has a strictly smaller color set than the head, and the goal is to minimize the total number of colors. We show that the problem is polynomially solvable in the class of triangle-free and Hamiltonian DAGs and also disclose the relationship of this problem with several other models of intersection representations of graphs and digraphs

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

Algorithmic Graph Theory

The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions

Hole의 관점에서 그래프와 유향그래프의 구조에 관한 연구

학위논문(박사)--서울대학교 대학원 :사범대학 수학교육과,2019. 8. 김서령.이 논문에서는 유향그래프와 그래프의 홀의 관점에서 계통발생 그래프와 그래프의 삼각화에 대하여 연구한다. 길이 4 이상인 유도된 싸이클을 홀이라 하고 홀이 없는 그래프를 삼각화된 그래프라 한다. 구체적으로, 싸이클을 갖지 않는 유향그래프의 계통발생 그래프가 삼각화된 그래프인지 판정하고, 주어진 그래프를 삼각화하여 클릭수가 크게 차이 나지 않는 그래프를 만드는 방법을 찾고자 한다. 이 논문은 연구 내용에 따라 두 부분으로 나뉜다. 먼저 $(1, i)$ 유향그래프와 $(i, 1)$ 유향그래프의 계통발생 그래프를 완전하게 특징화하고, $(2, j)$ 유향그래프 $D$의 모든 유향변에서 방향을 제거한 그래프가 삼각화된 그래프이면, $D$의 계통발생 그래프 역시 삼각화된 그래프임을 보였다. 또한 적은 수의 삼각형을 갖는 연결된 그래프의 계통발생수를 계산한 정리를 확장하여 많은 수의 삼각형을 포함한 연결된 그래프의 계통발생수를 계산하였다. 다른 한 편 그래프 $G$의 비삼각화 지수 $i(G)$에 대하여 $\omega(G^*)-\omega(G) \le i(G)$를 만족하는 $G$의 삼각화된 그래프 $G^*$가 존재함을 보였다. 그리고 이를 도구로 이용하여 NC property를 만족하는 그래프가 Hadwiger 추측과 Erd\H{o}s-Faber-Lov\'{a}sz 추측을 만족함을 증명하고, 비삼각화 지수가 유계인 그래프들이 linearly $\chi$-bounded임을 증명하였다.This thesis aims at studying phylogeny graphs and graph completions in the aspect of holes of graphs or digraphs. A hole of a graph is an induced cycle of length at least four and a graph is chordal if it does not contain a hole. Specifically, we determine whether the phylogeny graphs of acyclic digraphs are chordal or not and find a way of chordalizing a graph without increasing the size of maximum clique not so much. In this vein, the thesis is divided into two parts. In the first part, we completely characterize phylogeny graphs of $(1, i)$ digraphs and $(i,1)$ digraphs, respectively, for a positive integer $i$. Then, we show that the phylogeny graph of a $(2,j)$ digraph $D$ is chordal if the underlying graph of $D$ is chordal for any positive integer $j$. In addition, we extend the existing theorems computing phylogeny numbers of connected graph with a small number of triangles to results computing phylogeny numbers of connected graphs with many triangles. In the second part, we present a minimal chordal supergraph $G^*$ of a graph $G$ satisfying the inequality $\omega(G^*) - \omega(G) \le i(G)$ for the non-chordality index $i(G)$ of $G$. Using the above chordal supergraph as a tool, we prove that the family of graphs satisfying the NC property satisfies the Hadwiger conjecture and the Erd\H{o}s-Faber-Lov\'{a}sz Conjecture, and the family of graphs with bounded non-chordality indices is linearly $\chi$-bounded.Contents Abstract i 1 Introduction 1 1.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Phylogeny graphs . . . . . . . . . . . . . . . . . . . . . 8 1.2.2 Graph colorings and chordal completions . . . . . . . . 14 2 Phylogeny graphs 19 2.1 Chordal phylogeny graphs . . . . . . . . . . . . . . . . . . . . 19 2.1.1 (1,j) phylogeny graphs and (i,1) phylogeny graphs . . 20 2.1.2 (2,j) phylogeny graphs . . . . . . . . . . . . . . . . . . 28 2.2 The phylogeny number and the triangles and the diamonds of a graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3 A new minimal chordal completion 61 3.1 Graphs with the NC property . . . . . . . . . . . . . . . . . . 64 3.2 The Erd˝ os-Faber-Lovász Conjecture . . . . . . . . . . . . . . . 73 3.3 A minimal chordal completion of a graph . . . . . . . . . . . . 80 3.3.1 Non-chordality indices of graphs . . . . . . . . . . . . . 80 3.3.2 Making a local chordalization really local . . . . . . . . 89 3.4 New χ-bounded classes . . . . . . . . . . . . . . . . . . . . . . 97 Abstract (in Korean) 107Docto

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th