Related papers: A coarse Gallai theorem

In 1966, Gallai asked whether all longest paths in a connected graph share a common vertex. Counterexamples indicate that this is not true in general. However, Gallai's question is positive for certain well-known classes of connected…

For a graph $G$ and integer $k\geq1$, we define the token graph $F_k(G)$ to be the graph with vertex set all $k$-subsets of $V(G)$, where two vertices are adjacent in $F_k(G)$ whenever their symmetric difference is a pair of adjacent…

A minor-model of a graph $H$ in a graph $G$ is a subgraph of $G$ that can be contracted to $H$. We prove that for a positive integer $\ell$ and a non-empty planar graph $H$ with at least $\ell-1$ connected components, there exists a…

Gallai's path decomposition conjecture states that if $G$ is a connected graph on $n$ vertices, then the edges of $G$ can be decomposed into at most $\lceil \frac{n }{2} \rceil$ paths. A graph is said to be an odd semi-clique if it can be…

Let $G$ be a finite graph and $K[G]$ the edge ring of $G$. Based on the technique of Gr\"obner bases and initial ideals, it will be proved that, given integers $f$ and $d$ with $7 \leq f \leq d$, there exists a finite graph $G$ on…

In 1963, Corr\'adi and Hajnal proved that for all $k\geq1$ and $n\geq3k$, every graph $G$ on $n$ vertices with minimum degree $\delta(G)\geq2k$ contains $k$ disjoint cycles. The bound $\delta(G) \geq 2k$ is sharp. Here we characterize those…

We prove a number of results related to the computational complexity of recognizing well-covered graphs. Let $k$ and $s$ be positive integers and let $G$ be a graph. Then $G$ is said - $\mathbf{W_k}$ if for any $k$ pairwise disjoint…

A colored complete graph is said to be Gallai-colored if it contains no rainbow triangle. This property has been shown to be equivalent to the existence of a partition of the vertices (of every induced subgraph) in which at most two colors…

Let $D$ be a digraph. Given a set of vertices $S \subseteq V(D)$, an $S$-path partition $\mathcal{P}$ of $D$ is a collection of paths of $D$ such that $\{V(P) \colon P \in \mathcal{P}\}$ is a partition of $V(D)$ and $|V(P) \cap S| = 1$ for…

We prove a 1985 conjecture of Gy\'arf\'as that for all $k,\ell$, every graph with sufficiently large chromatic number contains either a complete subgraph with $k$ vertices or an induced cycle of length at least $\ell$.

A $1$-factor in an $n$-vertex graph $G$ is a collection of $\frac{n}{2}$ vertex-disjoint edges and a $1$-factorization of $G$ is a partition of its edges into edge-disjoint $1$-factors. Clearly, a $1$-factorization of $G$ cannot exist…

In this paper, we show that for given positive integer C, there are only finitely many distance-regular graphs with valency k at least three, diameter D at least six and k2/k<=C. This extends a conjecture of Bannai and Ito.

We call a finite undirected graph minimally k-matchable if it has at least k distinct perfect matchings but deleting any edge results in a graph which has not. An odd subdivision of some graph G is any graph obtained by replacing every edge…

A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai $k$-coloring is a Gallai coloring that uses at most $k$ colors. For an integer $k\geq 1$, the Gallai-Ramsey number $GR_k(H)$ of a given…

We introduce a common generalization of the strong Hanani-Tutte theorem and the weak Hanani-Tutte theorem: if a graph $G$ has a drawing $D$ in the plane where every pair of independent edges crosses an even number of times, then $G$ has a…

A separating path system for a graph $G$ is a collection $\mathcal{P}$ of paths in $G$ such that for every two edges $e$ and $f$ in $G$, there is a path in $\mathcal{P}$ that contains $e$ but not $f$. We show that every $n$-vertex graph has…

A directed graph (digraph) $ D $ is $ k $-linked if $ |D| \geq 2k $, and for any $ 2k $ distinct vertices $ x_1, \ldots, x_k, y_1, \ldots, y_k $ of $ D $, there exist vertex-disjoint paths $ P_1, \ldots, P_k $ such that $ P_i $ is a path…

Perhaps the very first elementary exercise one encounters in graph theory is the result that any graph on at least two vertices must have at least two vertices with the same degree. There are various ways in which this result can be…

Magnant and Martin conjectured that the vertex set of any $d$-regular graph $G$ on $n$ vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this…

Let G be a graph with set of vertices 1,...,n and adjacency matrix A of size nxn. Let d(i,j)=d, we say that f_d:N->N is a d-function on G if for every pair of vertices i,j and k>=d, we have a_ij^(k)=f_d(k). If this function f_d exists on G…