Related papers: Tuza's Conjecture for Threshold Graphs

A graph $ G $ is minimally $ t $-tough if the toughness of $ G $ is $ t $ and deletion of any edge from $ G $ decreases its toughness. Katona et al. conjectured that the minimum degree of any minimally $ t $-tough graph is $ \lceil 2t\rceil…

A connected graph $G$ with a perfect matching is said to be $k$-extendable for integers $k$, $1 \leq k\leq \frac{|V(G)|}{2}-1$, if any matching in $G$ of size $k$ is contained in a perfect matching of $G$. A $k$-extendable graph is minimal…

A graph is "$H$-free" if it has no induced subgraph isomorphic to $H$. A conjecture of Conlon, Fox and Sudakov states that for every graph $H$, there exists $s>0$ such that in every $H$-free graph with $n>1$ vertices, either some vertex has…

A conjecture of Richter and Salazar about graphs that are critical for a fixed crossing number $k$ is that they have bounded bandwidth. A weaker well-known conjecture of Richter is that their maximum degree is bounded in terms of $k$. In…

Every $n$-vertex planar triangle-free graph with maximum degree at most $3$ has an independent set of size at least $\frac{3}{8}n$. This was first conjectured by Albertson, Bollob\'as and Tucker, and was later proved by Heckman and Thomas.…

We show that for every graph without isolated edge, the edges can be assigned weights from {1,2,3} so that no two neighbors receive the same sum of incident edge weights. This solves a conjecture of Karo\'{n}ski, Luczak, and Thomason from…

A famous conjecture of Tuza states that the minimum number of edges needed to cover all the triangles in a graph is at most twice the maximum number of edge-disjoint triangles. This conjecture was couched in a broader setting by Aharoni and…

The distinguishing index of a simple graph $G$, denoted by $D'(G)$, is the least number of labels in an edge labeling of $G$ not preserved by any non-trivial automorphism. It was conjectured by Pil\'sniak (2015) that for any 2-connected…

Lovasz and Plummer conjectured that there exists a fixed positive constant c such that every cubic n-vertex graph with no cutedge has at least 2^(cn) perfect matchings. Their conjecture has been verified for bipartite graphs by Voorhoeve…

A long-standing conjecture of Berge suggests that every bridgeless cubic graph can be expressed as a union of at most five perfect matchings. This conjecture trivially holds for $3$-edge-colourable cubic graphs, but remains widely open for…

We study the typical structure and the number of triangle-free graphs with $n$ vertices and $m$ edges where $m$ is large enough so that a typical triangle-free graph has a cut containing nearly all of its edges, but may not be bipartite.…

In this paper, we study a conjecture of Andries E. Brouwer from 1996 regarding the minimum number of vertices of a strongly regular graph whose removal disconnects the graph into non-singleton components. We show that strongly regular…

We prove that any $n$-vertex graph whose complement is triangle-free contains $n^2/12-o(n^2)$ edge-disjoint triangles. This is tight for the disjoint union of two cliques of order $n/2$. We also prove a corresponding stability theorem, that…

In 1954, Tutte conjectured that every bridgeless graph has a nowhere-zero 5-flow. Let $\omega$ be the minimum number of odd cycles in a 2-factor of a bridgeless cubic graph. Tutte's conjecture is equivalent to its restriction to cubic…

A graph $G$ of order $n$ is said to be $k$-factor-critical for integers $1\leq k< n$, if the removal of any $k$ vertices results in a graph with a perfect matching. A $k$-factor-critical graph is minimal if for every edge, the deletion of…

A graph is $k$-planar if it can be drawn in the plane such that no edge is crossed more than $k$ times. While for $k=1$, optimal $1$-planar graphs, i.e., those with $n$ vertices and exactly $4n-8$ edges, have been completely characterized,…

There are many hard conjectures in graph theory, like Tutte's 5-flow conjecture, and the 5-cycle double cover conjecture, which would be true in general if they would be true for cubic graphs. Since most of them are trivially true for…

In this paper, we prove a tight minimum degree condition in general graphs for the existence of paths between two given endpoints, whose lengths form a long arithmetic progression with common difference one or two. This allows us to obtain…

Tutte conjectured in 1972 that every 4-edge connected graph has a nowhere-zero 3-flow. This has long been known to be equivalent to the conjecture that every 5-regular 4-edge-connected graph has an edge orientation in which every out-degree…

A graph is called $2K_2$-free if it does not contain two independent edges as an induced subgraph. Mou and Pasechnik conjectured that every $\frac{3}{2}$-tough $2K_2$-free graph with at least three vertices has a spanning trail with maximum…