arXiv subjects

Alexandr Kostochka

Publications and source records attributed to Alexandr Kostochka.

79 records · Page 5Linked to original sources

arXiv2013

Turan Problems and Shadows I: Paths and Cycles

A $k$-path is a hypergraph P_k = e_1,e_2,...,e_k such that |e_i \cap e_j| = 1 if |j - i| = 1 and e_i \cap e_j is empty otherwise. A k-cycle is a hypergraph C_k = e_1,e_2,.. ,e_k obtained from a (k-1)-path e_1,e_2,...,e_{k-1} by adding an edge e_k that shares one vertex with e_1, another vertex with e_{k-1} and is disjoint from the other edges. Let ex_r(n,G) be the maximum number of edges in an r-graph with n vertices not containing a given r-graph G. We determine ex_r(n, P_k) and ex_r(n, C_k) exactly for all k \ge 4 and r \ge 3 and $n$ sufficiently large and also characterize the extremal examples. The case k = 3 was settled by Frankl and Füredi. This work is the next step in a long line of research beginning with conjectures of Erd\H os and Sós from the early 1970's. In particular, we extend the work (and settle a recent conjecture) of Füredi, Jiang and Seiver who solved this problem for P_k when r \ge 4 and of Füredi and Jiang who solved it for C_k when r \ge 5. They used the delta system method, while we use a novel approach which involves random sampling from the shadow of an r-graph.

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arXiv2012

Ore's Conjecture for $k=4$ and Gr\" otzsch Theorem

A graph $G$ is $k$-{\em 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. In a very recent paper, we gave a lower bound, $f_k(n) \geq F(k,n)$, that is sharp for every $n=1\,({\rm mod}\, k-1)$. It is also sharp for $k=4$ and every $n\geq 6$. In this note, we present a simple proof of the bound for $k=4$. It implies the case $k=4$ of the conjecture by Ore from 1967 that for every $k\geq 4$ and $n\geq k+2$, $f_k(n+k-1)=f(n)+\frac{k-1}{2}(k - \frac{2}{k-1})$. We also show that our result implies a simple short proof of the Gr\" otzsch Theorem that every triangle-free planar graph is 3-colorable.

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arXiv2012

Ore's Conjecture on color-critical graphs is almost true

A graph $G$ 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. We give a lower bound, $f_k(n) \geq F(k,n)$, that is sharp for every $n=1 ({\rm mod} k-1)$. It is also sharp for $k=4$ and every $n\geq 6$. The result improves the classical bounds by Gallai and Dirac and subsequent bounds by Krivelevich and Kostochka and Stiebitz. It establishes the asymptotics of $f_k(n)$ for every fixed $k$. It also proves that the conjecture by Ore from 1967 that for every $k\geq 4$ and $n\geq k+2$, $f_k(n+k-1)=f(n)+\frac{k-1}{2}(k - \frac{2}{k-1})$ holds for each $k\geq 4$ for all but at most $k^3/12$ values of $n$. We give a polynomial-time algorithm for $(k-1)$-coloring a graph $G$ that satisfies $|E(G[W])| < F_k(|W|)$ for all $W \subseteq V(G)$, $|W| \geq k$. We also present some applications of the result.

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arXiv2012

Large rainbow matchings in large graphs

A \textit{rainbow subgraph} of an edge-colored graph is a subgraph whose edges have distinct colors. The \textit{color degree} of a vertex $v$ is the number of different colors on edges incident to $v$. We show that if $n$ is large enough (namely, $n\geq 4.25k^2$), then each $n$-vertex graph $G$ with minimum color degree at least $k$ contains a rainbow matching of size at least $k$.

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arXiv2012

Harmonious Coloring of Trees with Large Maximum Degree

A harmonious coloring of $G$ is a proper vertex coloring of $G$ such that every pair of colors appears on at most one pair of adjacent vertices. The harmonious chromatic number of $G$, $h(G)$, is the minimum number of colors needed for a harmonious coloring of $G$. We show that if $T$ is a forest of order $n$ with maximum degree $Δ(T)\geq \frac{n+2}{3}$, then $$h(T)= Δ(T)+2, & if $T$ has non-adjacent vertices of degree $Δ(T)$; Δ(T)+1, & otherwise. $$ Moreover, the proof yields a polynomial-time algorithm for an optimal harmonious coloring of such a forest.

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arXiv2011

Choosability with separation of complete multipartite graphs and hypergraphs

For a hypergraph G and a positive integer s, let χ_{\ell} (G,s) be the minimum value of l such that G is L-colorable from every list L with |L(v)|=l for each v\in V(G) and |L(u)\cap L(v)|\leq s for all u, v\in e\in E(G). This parameter was studied by Kratochvíl, Tuza and Voigt for various kinds of graphs. Using randomized constructions we find the asymptotics of χ_{\ell} (G,s) for balanced complete multipartite graphs and for complete k-partite k-uniform hypergraphs.

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arXiv2007

Hadwiger Number and the Cartesian Product Of Graphs

The Hadwiger number mr(G) of a graph G is the largest integer n for which the complete graph K_n on n vertices is a minor of G. Hadwiger conjectured that for every graph G, mr(G) >= chi(G), where chi(G) is the chromatic number of G. In this paper, we study the Hadwiger number of the Cartesian product G [] H of graphs. As the main result of this paper, we prove that mr(G_1 [] G_2) >= h\sqrt{l}(1 - o(1)) for any two graphs G_1 and G_2 with mr(G_1) = h and mr(G_2) = l. We show that the above lower bound is asymptotically best possible. This asymptotically settles a question of Z. Miller (1978). As consequences of our main result, we show the following: 1. Let G be a connected graph. Let the (unique) prime factorization of G be given by G_1 [] G_2 [] ... [] G_k. Then G satisfies Hadwiger's conjecture if k >= 2.log(log(chi(G))) + c', where c' is a constant. This improves the 2.log(chi(G))+3 bound of Chandran and Sivadasan. 2. Let G_1 and G_2 be two graphs such that chi(G_1) >= chi(G_2) >= c.log^{1.5}(chi(G_1)), where c is a constant. Then G_1 [] G_2 satisfies Hadwiger's conjecture. 3. Hadwiger's conjecture is true for G^d (Cartesian product of G taken d times) for every graph G and every d >= 2. This settles a question by Chandran and Sivadasan (They had shown that the Hadiwger's conjecture is true for G^d if d >= 3.)

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