Related papers: On maximum graphs in Tutte polynomial posets

For a graph $G=(V,E)$, let $\tau(G)$ denote the minimum number of pairwise edge disjoint complete bipartite subgraphs of $G$ so that each edge of $G$ belongs to exactly one of them. It is easy to see that for every graph $G$, $\tau(G) \leq…

Popielarz, Sahasrabudhe and Snyder in 2018 proved that maximal $K_{r+1}$-free graphs with $(1-\frac{1}{r})\frac{n^2}{2}-o(n^{\frac{r+1}{r}})$ edges contain a complete $r$-partite subgraph on $n-o(n)$ vertices. This was very recently…

The poset of maximal tubings of a graph generalizes several well-known and remarkable partial orders. Notable examples include the weak Bruhat order and the Tamari lattice, posets of maximal tubings for the complete graph and the path…

A graph $G$ is total weight $(k,k')$-choosable if for any total list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real numbers, and each edge $e$ a set $L(e)$ of $k'$ real numbers, there is a proper total…

Perfect graphs can be described as the graphs whose stable set polytopes are defined by their non-negativity and clique inequalities (including edge inequalities). In 1975, Chv\'{a}tal defined an analogous class of t-perfect graphs, which…

Lov\'asz and Cherkassky discovered in the 1970s independently that if $ G $ is a finite graph with a given set $ T $ of terminal vertices such that $ G $ is inner Eulerian, then the maximal number of edge-disjoint paths connecting distinct…

A graph $G$ is $[a,b]$-covered if for each edge $e$ of $G$ there is an $[a,b]$-factor containing it. For $a=b=1$, an $[a,b]$-covered graph is a matching covered graph. The structural theory of matching covered graphs constitutes a…

Let $\mathbf G$ be a graphing, that is a Borel graph defined by $d$ measure preserving involutions. We prove that if $\mathbf G$ is {\em treeable} then it arises as the local limit of some sequence $(G_n)_{n\in\mathbb{N}}$ of graphs with…

In this paper we further investigate the well-studied problem of finding a perfect matching in a regular bipartite graph. The first non-trivial algorithm, with running time $O(mn)$, dates back to K\"{o}nig's work in 1916 (here $m=nd$ is the…

Kostochka and Yancey proved that every 5-critical graph G satisfies: |E(G)|>= (9/4)|V(G)| - 5/4. A construction of Ore gives an infinite family of graphs meeting this bound. We prove that there exists e,d > 0 such that if G is a 5-critical…

We introduce the notions of arithmetic colorings and arithmetic flows over a graph with labelled edges, which generalize the notions of colorings and flows over a graph. We show that the corresponding arithmetic chromatic polynomial and…

A fundamental result in extremal graph theory is attributed to Mantel's theorem, which states that every graph on $n$ vertices with more than $\lfloor n^2/4 \rfloor$ edges must contain a triangle. Lov\'{a}sz and Simonovits (1975) provided a…

The Tutte polynomial of a graph or a matroid, named after W. T. Tutte, has the important universal property that essentially any multiplicative graph or network invariant with a deletion and contraction reduction must be an evaluation of…

The maximum number of vertices in a graph of maximum degree $\Delta\ge 3$ and fixed diameter $k\ge 2$ is upper bounded by $(1+o(1))(\Delta-1)^{k}$. If we restrict our graphs to certain classes, better upper bounds are known. For instance,…

Let $m:=|E(G)|$ sufficiently large and $s:=(m-1)/3$. We show that unless the maximum degree $\Delta > 2s$, there is a weighting $w:E\cup V\to \{0,1,...,s\}$ so that $w(uv)+w(u)+w(v)\ne w(u'v')+w(u')+w(v')$ whenever $uv\ne u'v'$ (such a…

We investigate the extremal properties of saturated partial plane embeddings of maximal planar graphs. For a planar graph $G$, the plane-saturation number $\mathrm{sat}_{\mathcal{P}}(G)$ denotes the minimum number of edges in a plane…

A soft random graph $G(n,r,p)$ can be obtained from the random geometric graph $G(n,r)$ by keeping every edge in $G(n,r)$ with probability $p$. The soft random simplicial complexes is a model for random simplicial complexes built over the…

A {\it tiered graph} $G=(V,E)$ with $m $ tiers is a simple graph with $V\subseteq \brk{n}$, where $\brk{n}=\{1,2,\cdots,n\}$, and with a surjective map $t$ from $V$ to $\brk{m}$ such that if $v$ is a vertex adjacent to $v'$ in $G$ with…

A proper total weighting of a graph $G$ is a mapping $\phi$ which assigns to each vertex and each edge of $G$ a real number as its weight so that for any edge $uv$ of $G$, $\sum_{e \in E(v)}\phi(e)+\phi(v) \ne \sum_{e \in…

A solution to a problem of Erd\H{o}s, Rubin and Taylor is obtained by showing that if a graph $G$ is $(a:b)$-choosable, and $c/d > a/b$, then $G$ is not necessarily $(c:d)$-choosable. The simplest case of another problem, stated by the same…