Related papers: Between 2- and 3-colorability

We study random graphs, both $G(n,p)$ and $G(n,m)$, with random orientations on the edges. For three fixed distinct vertices s,a,b we study the correlation, in the combined probability space, of the events a -> s and s -> b. For G(n,p), we…

We prove that there exists a positive constant \epsilon such that if \log n / n \le p \le n^{-1+\epsilon}, then asymptotically almost surely the random graph G ~ G(n,p) contains a collection of \lfloor \delta(G)/2 \rfloor edge-disjoint…

Let $G = (V, E)$ be an $n$-vertex edge-colored graph. In 2013, H. Li proved that if every vertex $v \in V$ is incident to at least $(n+1)/2$ distinctly colored edges, then $G$ admits a rainbow triangle. We prove that the same hypothesis…

The chromatic number of an planar graph is not greater than four and this is known by the famous four color theorem and is equal to two when the planar graph is bipartite. When the planar graph is even-triangulated or all cycles are greater…

A {\em hole} is an induced cycle of length at least 4, a $k$-hole is a hole of length $k$, and an {\em odd hole} is a hole of odd length. Let $\ell\ge 2$ be an integer. Let ${\cal A}_{\ell}$ be the family of graphs of girth at least $2\ell$…

We study the following Steinberg-type problem on circular coloring: for an odd integer $k\ge 3$, what is the smallest number $f(k)$ such that every planar graph of girth $k$ without cycles of length from $k+1$ to $f(k)$ admits a…

In this paper we study cycles in random bipartite graph $G(n,n,p)$. We prove that if $p\gg n^{-2/3}$, then $G(n,n,p)$ a.a.s. satisfies the following. Every subgraph $G'\subset G(n,n,p)$ with more than $(1+o(1))n^2p/2$ edges contains a cycle…

This paper concerns rigidity of the mapping class groups. We show that any homomorphism $\phi:{\rm Mod}_g\to {\rm Mod}_h$ between mapping class groups of closed orientable surfaces with distinct genera $g>h$ is trivial if $g\geq 3$ and has…

A cycle with four blocks $C(k_{1}, k_{2},k_{3},k_{4})$ is an oriented cycle formed of four blocks of lengths $k_{1}, k_{2}, k_{3}$ and $k_{4}$ respectively. Recently, Cohen et al. conjectured that for every positive integers $k_{1}, k_{2},…

The interplay of minimum degree conditions and structural properties of large graphs with forbidden subgraphs is a central topic in extremal graph theory. For a given graph $F$ we define the homomorphism threshold as the infimum over all…

Li, Nikiforov and Schelp conjectured that a 2-edge coloured graph G with order n and minimal degree strictly greater than 3n/4 contains a monochromatic cycle of length l, for all l at least four and at most n/2. We prove this conjecture for…

An $(m, n)$-colored-mixed graph $G=(V, A_1, A_2,\cdots, A_m, E_1, E_2,\cdots, E_n)$ is a graph having $m$ colors of arcs and $n$ colors of edges. We do not allow two arcs or edges to have the same endpoints. A homomorphism from an…

Given an $n$ vertex graph whose edges have colored from one of $r$ colors $C=\{c_1,c_2,\ldots,c_r\}$, we define the Hamilton cycle color profile $hcp(G)$ to be the set of vectors $(m_1,m_2,\ldots,m_r)\in [0,n]^r$ such that there exists a…

If eps > 0 and p >= n^{-1/2 + eps}, in a binomial random graph G(n,p) a.a.s. the set of cycles which can be constructed as a symmetric difference of Hamilton circuits is as large as parity by itself permits (all cycles if n is odd, all even…

Let G be a graph. It was proved that if G is a planar graph without {4, 6, 7}-cycles and without two 5-cycles sharing exactly one edge, then G 3-colorable. We observed that the proof of this result is not correct.

To any two graphs G and H one can associate a cell complex Hom(G,H) by taking all graph multihomorphisms from G to H as cells. In this paper we prove the Lovasz Conjecture which states that if Hom(C_{2r+1},G) is k-connected, then…

We classify nonconstant morphisms $\mathbb{P}^m \to G/P$ for $m \le 4$ when $G = SL(n,\mathbb{C})$ (type~$A$) for a minimal parabolic subgroup $P$. Using the Borel presentation of cohomology and explicit Schubert intersection identities, we…

A signed bipartite (simple) graph $(G, \sigma)$ is said to be $C_{-4}$-critical if it admits no homomorphism to $C_{-4}$ (a negative 4-cycle) but every proper subgraph of it does. In this work, first of all we show that the notion of…

Let $\mathcal A$ and $\mathcal B$ be two (complex) algebras. A linear map $\phi:{\mathcal A}\to{\mathcal B}$ is called $n$-homomorphism if $\phi(a_{1}... a_{n})=\phi(a_{1})...\phi(a_{n})$ for each $a_{1},...,a_{n}\in{\mathcal A}.$ In this…

For any two graphs $G$ and $H$ Lov\'asz has defined a cell complex $Hom(G,H)$ having in mind the general program that the algebraic invariants of these complexes should provide obstructions to graph colorings. Here we announce the proof of…