Related papers: Graphes parfaits : structure et algorithmes
A total graph is an ordered triple $(V_0, V_1, E)$, where $V_0, V_1$ are the sets of empty and full vertices, respectively, $V_0 \cap V_1 = \emptyset$, and the set of edges $E$ is a subset of \(\binom{V_0 \cup V_1}{2}\) $(E\cap(V_0 \cup…
A proper edge colouring of a graph is adjacent vertex distinguishing if no two adjacent vertices see the same set of colours. Using a clever application of the Local Lemma, Hatami (2005) proved that every graph with maximum degree $\Delta$…
This paper explores the application of a new algebraic method of color exchanges to the edge coloring of simple graphs. Vizing's theorem states that the edge coloring of a simple graph $G$ requires either $\Delta$ or $\Delta+1$ colors,…
We define a perfect coloring of a graph $G$ as a proper coloring of $G$ such that every connected induced subgraph $H$ of $G$ uses exactly $\omega(H)$ many colors where $\omega(H)$ is the clique number of $H$. A graph is perfectly colorable…
A strong edge-coloring of a graph $G$ is a coloring of the edges such that every color class induces a matching in $G$. The strong chromatic index of a graph is the minimum number of colors needed in a strong edge-coloring of the graph. In…
A vertex colouring of some graph is called perfect if each vertex of colour $i$ has exactly $a_{ij}$ neighbours of colour $j$. Being perfect imposes several restrictions on the colour incidence matrix $(a_{ij})$. We list several (old and…
In 1994, it was conjectured by Fan and Raspaud that every simple bridgeless cubic graph has three perfect matchings whose intersection is empty. In this paper we answer a question recently proposed by Mkrtchyan and Vardanyan, by giving an…
Given a graphic degree sequence $D$, let $\chi(D)$ (respectively $\omega(D)$, $h(D)$, and $H(D)$) denote the maximum value of the chromatic number (respectively, the size of the largest clique, largest clique subdivision, and largest clique…
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 study the perfect matching reconfiguration problem: Given two perfect matchings of a graph, is there a sequence of flip operations that transforms one into the other? Here, a flip operation exchanges the edges in an alternating cycle of…
We continue research into a well-studied family of problems that ask whether the vertices of a graph can be partitioned into sets $A$ and~$B$, where $A$ is an independent set and $B$ induces a graph from some specified graph class ${\cal…
Let $G$ be a graph of order $n$, maximum degree at most $\Delta$, and no component of order $2$. Inspired by the famous 1-2-3-conjecture, Bensmail, Marcille, and Orenga define a proper pushing scheme of $G$ as a function…
It is consistent that for every monotonically increasing function f:omega->omega there is a graph with size and chromatic number aleph_1 in which every n-chromatic subgraph has at least f(n) elements (n >= 3). This solves a $250 problem of…
The total Betti number of the independence complex of a graph is an intriguing graph invariant. Kalai and Meshulam have raised the question on its relation to cycles and the chromatic number of a graph, and a recent conjecture on that theme…
A coloring of vertices of a graph is called perfect if, for every vertex, the collection of colors of its neighbors depends only on its own color. The correspondent color partition of vertices is called equitable. We note that a number of…
A vertex colouring of some graph is called perfect if each vertex of colour $i$ has the same number $a_{ij}$ of neighbours of colour $j$. Here we determine all perfect colourings of the edge graphs of the hypercube in dimensions 4 and 5 by…
A hole is a chordless cycle with at least four vertices. A hole is odd if it has an odd number of vertices. A dart is a graph which vertices $a, b, c, d, e$ and edges $ab, bc, bd, be, cd, de$. Dart-free graphs have been actively studied in…
In this paper, we study orthogonal colourings of random geometric graphs. Two colourings of a graph are orthogonal if they have the property that when two vertices receive the same colour in one colouring, then those vertices receive…
Counting perfect matchings has played a central role in the theory of counting problems. The permanent, corresponding to bipartite graphs, was shown to be #P-complete to compute exactly by Valiant (1979), and a fully polynomial randomized…
We study the class of simple graphs $\mathcal{G}^*$ for which every pair of distinct odd cycles intersect in at most one edge. We give a structural characterization of the graphs in $\mathcal{G}^*$ and prove that every $G \in \mathcal{G}^*$…