Related papers: Perfect graphs: a survey
We develop a theory of graph algebras over general fields. This is modeled after the theory developed by Freedman, Lov\'asz and Schrijver in [22] for connection matrices, in the study of graph homomorphism functions over real edge weight…
In this paper, we prove that given a 2-edge-coloured complete graph $K_{4n}$ that has the same number of edges of each colour, we can always find a perfect matching with an equal number of edges of each colour. This solves a problem posed…
We prove that there exist perfect graphs of arbitrarily large clique-chromatic number. These graphs can be obtained from cobipartite graphs by repeatedly gluing along cliques. This negatively answers a question raised by Duffus, Sands,…
Reed showed that, if two graphs are $P_4$-isomorphic, then either both are perfect or none of them is. In this note we will derive an analogous result for perfect digraphs.
Various classes of induced subgraphs are involved in the deepest results of graph theory and graph algorithms. A prominent example concerns the {\em perfection} of $G$ that the chromatic number of each induced subgraph $H$ of $G$ equals the…
We find the Ramsey number of a cycle vs. a complete graph when the order of the cycle is at least 4 times as large as the order of the complete graph. This partially confirms a conjecture of Erd\H{o}s, Faudree, Rousseau, and Schelp made in…
A vertex coloring of a given simple graph $G=(V,E)$ with $k$ colors ($k$-coloring) is a map from its vertex set to the set of integers $\{1,2,3,\dots, k\}$. A coloring is called perfect if the multiset of colors appearing on the neighbours…
We introduce the concept of matching connectivity as a notion of connectivity in graph admitting perfect matchings which heavily relies on the structural properties of those matchings. We generalise a result of Robertson, Seymour and Thomas…
We generalize the idea of cofinite groups, due to B. Hartley. First we define cofinite spaces in general. Then, as a special situation, we study cofinite graphs and their uniform completions. The idea of constructing a cofinite graph starts…
A coloring of vertices of a given graph is called perfect if the color structure of each ball of radius $1$ in the graph depends only on the color of the ball center. Let $n$ be a positive integer. We consider a lexicographic product of the…
In 1985, Golumbic and Scheinerman established an equivalence between comparability graphs and containment graphs, graphs whose vertices represent sets, with edges indicating set containment. A few years earlier, McMorris and Zaslavsky…
A spanning subgraph $F$ of a graph $G$ is called perfect if $F$ is a forest, the degree $d_F(x)$ of each vertex $x$ in $F$ is odd, and each tree of $F$ is an induced subgraph of $G$. We provide a short proof of the following theorem of A.D.…
We classify the finite quasisimple groups whose commuting graph is perfect and we give a general structure theorem for finite groups whose commuting graph is perfect.
Efficiently finding the maximum a posteriori (MAP) configuration of a graphical model is an important problem which is often implemented using message passing algorithms. The optimality of such algorithms is only well established for…
In a directed graph, a kernel is a subset of vertices that is both stable and absorbing. Not all digraphs have a kernel, but a theorem due to Boros and Gurvich guarantees the existence of a kernel in every clique-acyclic orientation of a…
Hypergraphs were introduced in 1973 by Berg\'e. This review aims at giving some hints on the main results that we can find in the literature, both on the mathematical side and on their practical usage. Particularly, different definitions of…
In 2006 Qian [J. Qian, Degree complete graphs; Discrete Mathematics 306 (2006), 533--537] introduced the concept of degree complete graphs for labeled graphs. He also gave a characterization of these graphs in terms of two forbidden…
We study the existence of perfect matchings in suitably chosen induced subgraphs of random biregular bipartite graphs. We prove a result similar to a classical theorem of Erdos and Renyi about perfect matchings in random bipartite graphs.…
We show that deciding whether a given graph $G$ of size $m$ has a unique perfect matching as well as finding that matching, if it exists, can be done in time $O(m)$ if $G$ is either a cograph, or a split graph, or an interval graph, or…
The concept of super line graph was introduced in the year 1995 by Bagga, Beineke and Varma. Given a graph with at least $r$ edges, the super line graph of index $r$, $L_r(G)$, has as its vertices the sets of $r$ edges of $G$, with two…