# Search results for: Alexandr Kostochka

Journal of Graph Theory > 100 > 3 > 578 - 607

*connected*if all the edges of $M$ are in the same component of $G$. Following Łuczak, there have been many results using the existence of large connected matchings in cluster graphs with respect to regular partitions of large graphs to show the existence of long paths and other structures in these graphs. We prove exact Ramsey‐type bounds on the...

Journal of Graph Theory > 99 > 4 > 758 - 782

Journal of Graph Theory > 93 > 2 > 203 - 221

Graphs and Combinatorics > 2019 > 35 > 6 > 1495-1502

*union-closed*if it contains the union of any two sets in it. The

*Union-Closed Sets Conjecture*of Frankl from 1979 states that each union-closed family contains an element that belongs to at least half of the members of the family. In this paper, we study structural properties of union-closed families. It is known that under the inclusion relation, every union-closed...

Graphs and Combinatorics > 2019 > 35 > 2 > 513-537

*packing*

*k*

*-coloring*of a graph

*G*is a partition of

*V*(

*G*) into sets $$V_1,\ldots ,V_k$$ V 1 , … , V k such that for each $$1\le i\le k$$ 1 ≤ i ≤ k the distance between any two distinct $$x,y\in V_i$$ x , y ∈ V i is at least $$i+1$$ i + 1 . The

*packing chromatic number*, $$\chi _p(G)$$ χ p ( G ) , of a graph

*G*is the minimum

*k*such that

*G*has a packing...

Journal of Graph Theory > 89 > 2 > 176 - 193

*n*‐vertex nonhamiltonian graph

*G*with minimum degree $\delta \left(G\right)\ge d$ has at most $h(n,d)$ edges. He also provides a sharpness example ${H}_{n,d}$ for all such pairs $n,d$. Previously, we showed a stability version of this result: for

*n*large enough, every nonhamiltonian graph

*G*on

*n*vertices with $\delta \left(G\right)\ge d$ and more than $h(n,d+$...

Journal of Graph Theory > 88 > 3 > 521 - 546

*Correspondence coloring*, or

*DP‐coloring*, is a generalization of list coloring introduced recently by Dvořák and Postle [11]. In this article, we establish a version of Dirac's theorem on the minimum number of edges in critical graphs [9] in the framework of DP‐colorings. A corollary of our main result answers a question posed by Kostochka and Stiebitz [15] on classifying list‐critical graphs that...

Combinatorica > 2018 > 38 > 4 > 887-934

*G*is

*k-critical*if it has chromatic number

*k*, but every proper subgraph of

*G*is (

*k*-1)-colorable. Let

*f*

_{k}(

*n*) denote the minimum number of edges in an

*n*-vertex

*k*-critical graph. Recently the authors gave a lower bound, $${f_k}\left( n \right) \geqslant \left\lceil {\frac{{\left( {k + 1} \right)\left( {k - 2} \right)\left| {V\left( G \right)} \right| - k\left[ {k - 3} \right]}}{{2\left( {k -...

Discrete Mathematics > 2018 > 341 > 2 > 474-483

Discrete Mathematics > 2017 > 340 > 11 > 2688-2690

European Journal of Combinatorics > 2017 > 65 > C > 122-129

Journal of Combinatorial Theory, Series B > 2017 > 122 > C > 457-478

Journal of Combinatorial Theory, Series B > 2016 > 121 > C > 197-228

Discrete Mathematics > 2016 > 339 > 8 > 2178-2185

Combinatorica > 2017 > 37 > 3 > 481-494

*G*of chromatic number

*k*≥

*k*

_{0}(

*ε*) contains cycles of at least

*k*

^{2−ε}different lengths as

*k*→∞. In this paper, we prove the stronger fact that every triangle-free graph

*G*of chromatic number

*k*≥

*k*

_{0}(

*ε*) contains cycles of 1/64(1 −

*ε*)

*k*

^{2}log

*k*/4 consecutive lengths, and a cycle of length at least 1/4(1 −

*ε*)

*k*

^{2}log

*k*. As there exist triangle-free...

Journal of Graph Theory > 81 > 4 > 403 - 413

*G*is $(j,k)$‐

*colorable*if $V\left(G\right)$ can be partitioned into two sets ${V}_{j}$ and ${V}_{k}$ so that the maximum degree of $G\left[{V}_{j}\right]$ is at most

*j*and of $G\left[{V}_{k}\right]$ is at most

*k*. While the problem of verifying whether a graph is (0, 0)‐colorable is easy, the similar problem with $(j,k)$ in place of (0, 0) is NP‐complete for all nonnegative

*j*and

*k*with $j+k\ge 1$. Let ${F}_{j,k}\left(g\right)$ denote the supremum of all

*x*such that for some constant...

Discrete Mathematics > 2016 > 339 > 2 > 682-688

Israel Journal of Mathematics > 2016 > 214 > 1 > 315-331

*r*-augmented tree is a rooted tree plus

*r*edges added from each leaf to ancestors. For

*d, g, r*∈ N, we construct a bipartite

*r*-augmented complete

*d*-ary tree having girth at least

*g*. The height of such trees must grow extremely rapidly in terms of the girth. Using the resulting graphs, we construct sparse non-

*k*-choosable bipartite graphs, showing that maximum average degree at most 2(

*k*- 1)...

Journal of Combinatorial Theory, Series A > 2015 > 129 > Complete > 57-79

European Journal of Combinatorics > 2014 > 42 > Complete > 26-48