Related papers: Maxmaxflow and counting subgraphs

A graph homomorphism is a vertex map which carries edges from a source graph to edges in a target graph. The instances of the Weighted Maximum H-Colourable Subgraph problem (MAX H-COL) are edge-weighted graphs G and the objective is to find…

Extending the idea from the recent paper by Carbonero, Hompe, Moore, and Spirkl, for every function $f\colon\mathbb{N}\to\mathbb{N}\cup\{\infty\}$ with $f(1)=1$ and $f(n)\geq\binom{3n+1}{3}$, we construct a hereditary class of graphs…

We provide a finite forbidden induced subgraph characterization for the graph class $\varUpsilon_k$, for all $k \in \mathbb{N}_0$, which is defined as follows. A graph is in $\varUpsilon_k$ if for any induced subgraph, $\Delta \leq \chi -1…

Let $\chi'_\subset(G)$ be the least number of colours necessary to properly colour the edges of a graph $G$ with minimum degree $\delta\geq 2$ so that the set of colours incident with any vertex is not contained in a set of colours incident…

We introduce the notion of a generalized flow on a graph with coefficients in a R-representation and show that the module of flows is isomorphic to the first derived functor of the colimit. We generalize Kirchhoff's laws and build an exact…

The Hall ratio of a graph G is the maximum of |V(H)|/alpha(H) over all subgraphs H of G. Clearly, the Hall ratio of a graph is a lower bound for the fractional chromatic number. It has been asked whether conversely, the fractional chromatic…

A coloring of a graph is an assignment of colors to its vertices such that adjacent vertices have different colors. Two colorings are equivalent if they induce the same partition of the vertex set into color classes. Let $\mathcal{A}(G)$ be…

The boxicity (respectively cubicity) of a graph $G$ is the minimum non-negative integer $k$, such that $G$ can be represented as an intersection graph of axis-parallel $k$-dimensional boxes (respectively $k$-dimensional unit cubes) and is…

We prove that for any graph $G$ the (complex) zeros of its chromatic polynomial, $\chi_G(x)$, lie inside the disk centered at $0$ of radius $4.25 \Delta(G)$, where $\Delta(G)$ denotes the maximum degree of $G$. This improves on a recent…

A graph (digraph) $G=(V,E)$ with a set $T\subseteq V$ of terminals is called inner Eulerian if each nonterminal node $v$ has even degree (resp. the numbers of edges entering and leaving $v$ are equal). Cherkassky and Lov\'asz showed that…

Given a directed graph $G = (V, E)$ with $n$ vertices, $m$ edges and a designated source vertex $s\in V$, we consider the question of finding a sparse subgraph $H$ of $G$ that preserves the flow from $s$ up to a given threshold $\lambda$…

An \emph{interval $t$-coloring} of a multigraph $G$ is a proper edge coloring with colors $1,\dots,t$ such that the colors on the edges incident to every vertex of $G$ are colored by consecutive colors. A \emph{cyclic interval $t$-coloring}…

Although the chromatic number of a graph is not known in general, attempts have been made to find good bounds for the number. Here we prove that for a graph G with two forbidden subgraphs and maximum degree less than or equal to 2{\omega} -…

Graph polynomials are graph parameters invariant under graph isomorphisms which take values in a polynomial ring with a fixed finite number of indeterminates. We study graph polynomials from a model theoretic point of view. In this paper we…

Given an undirected graph $G$, we define a new object $H_G$, called the mp-chart of $G$, in the max-plus algebra. We use it, together with the max-plus permanent, to describe the complexity of graphs. We show how to compute the mean and the…

Let $G$ be an embedded graph and $A$ an edge subset of $G$. The partial dual of $G$ with respect to $A$, denoted by $G^A$, can be viewed as the geometric dual $G^*$ of $G$ over $A$. If $A=E(G)$, then $G^A=G^*$. Denote by $\gamma(G^A)$ the…

For a graph G, M(G) denotes the maximum multiplicity occurring of an eigenvalue of a symmetric matrix whose zero-nonzero pattern is given by edges of G. We introduce two combinatorial graph parameters T^-(G) and T^+(G) that give a lower and…

The \emph{vitality} of an arc/node of a graph with respect to the maximum flow between two fixed nodes $s$ and $t$ is defined as the reduction of the maximum flow caused by the removal of that arc/node. In this paper we address the issue of…

A graph with at most two vertices of the same degree is called antiregular (Merris 2003), maximally nonregular (Zykov 1990) or quasiperfect (Behzad, Chartrand 1967). If s_{k} is the number of independent sets of cardinality k in a graph G,…

Many well-studied problems in extremal combinatorics deal with the maximum possible size of a family of objects in which every pair of objects satisfies a given restriction. One problem of this type was recently raised by Alon, Gujgiczer,…