arXiv subjects

Maria Axenovich

Publications and source records attributed to Maria Axenovich.

At least 73 records · Page 4Linked to original sources

arXiv2016

On the strong chromatic number of graphs

The strong chromatic number, $χ_S(G)$, of an $n$-vertex graph $G$ is the smallest number $k$ such that after adding $k\lceil n/k\rceil-n$ isolated vertices to $G$ and considering {\bf any} partition of the vertices of the resulting graph into disjoint subsets $V_1, \ldots, V_{\lceil n/k\rceil}$ of size $k$ each, one can find a proper $k$-vertex-coloring of the graph such that each part $V_i$, $i=1, \ldots, \lceil n/k\rceil$, contains exactly one vertex of each color. For any graph $G$ with maximum degree $Δ$, it is easy to see that $χ_S(G)\geqΔ+1$. Recently, Haxell proved that $χ_S(G) \leq 3Δ-1$. In this paper, we improve this bound for graphs with large maximum degree. We show that $χ_S(G)\leq 2Δ$ if $Δ\geq n/6$ and prove that this bound is sharp.

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arXiv2016

Avoiding patterns in matrices via a small number of changes

Let ${\cal A}=\{A_1,\ldots, A_r\}$ be a partition of a set $\{1,\ldots,m\}\times\{1,\ldots, n\}$ into $r$ nonempty subsets, and $A=(a_{ij})$ be an $m\times n$ matrix. We say that $A$ has a pattern ${\cal A}$ provided that $a_{ij}=a_{i'j'}$ if and only if $(i,j),(i',j')\in A_t$ for some $t\in\{1,\ldots,r\}$. In this note we study the following function $f$ defined on the set of all $m\times n$ matrices $M$ with $s$ distinct entries: $f(M; {\cal A})$ is the smallest number of positions where the entries of $M$ need to be changed such that the resulting matrix does not have any submatrix with pattern ${\cal A}$. We give an asymptotically tight value for $$ f(m,n; s, {\cal A}) = \max\{f(M; {\cal A}): M \mbox{ is an } m\times n\mbox{ matrix with at most } s \mbox{ distinct entries}\} . $$

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arXiv2016

A version of Szemerédi's regularity lemma for multicolored graphs and directed graphs that is suitable for induced graphs

In this manuscript we develop a version of Szemerédi's regularity lemma that is suitable for analyzing multicolorings of complete graphs and directed graphs. In this, we follow the proof of Alon, Fischer, Krivelevich and M. Szegedy [Combinatorica, 20(4) (2000), 451--476] who prove a similar result for graphs. The purpose is to extend classical results on dense hereditary properties, such as the speed of the property or edit distance, to the above-mentioned combinatorial objects.

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arXiv2016

On the editing distance of graphs

An edge-operation on a graph $G$ is defined to be either the deletion of an existing edge or the addition of a nonexisting edge. Given a family of graphs $\mathcal{G}$, the editing distance from $G$ to $\mathcal{G}$ is the smallest number of edge-operations needed to modify $G$ into a graph from $\mathcal{G}$. In this paper, we fix a graph $H$ and consider ${\rm Forb}(n,H)$, the set of all graphs on $n$ vertices that have no induced copy of $H$. We provide bounds for the maximum over all $n$-vertex graphs $G$ of the editing distance from $G$ to ${\rm Forb}(n,H)$, using an invariant we call the {\it binary chromatic number} of the graph $H$. We give asymptotically tight bounds for that distance when $H$ is self-complementary and exact results for several small graphs $H$.

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arXiv2016

$Q_2$-free families in the Boolean lattice

For a family $\mathcal{F}$ of subsets of [n]=\{1, 2, ..., n} ordered by inclusion, and a partially ordered set P, we say that $\mathcal{F}$ is P-free if it does not contain a subposet isomorphic to P. Let $ex(n, P)$ be the largest size of a P-free family of subsets of [n]. Let $Q_2$ be the poset with distinct elements a, b, c, d, a<b, c<d; i.e., the 2-dimensional Boolean lattice. We show that $2N -o(N) \leq ex(n, Q_2)\leq 2.283261N +o(N), $ where $N = \binom{n}{\lfloor n/2 \rfloor}$. We also prove that the largest $Q_2$-free family of subsets of [n] having at most three different sizes has at most 2.20711N members.

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arXiv2016

Multicolor and directed edit distance

The editing of a combinatorial object is the alteration of some of its elements such that the resulting object satisfies a certain fixed property. The edit problem for graphs, when the edges are added or deleted, was first studied independently by the authors and Kézdy [J. Graph Theory (2008), 58(2), 123--138] and by Alon and Stav [Random Structures Algorithms (2008), 33(1), 87--104]. In this paper, a generalization of graph editing is considered for multicolorings of the complete graph as well as for directed graphs. Specifically, the number of edge-recolorings sufficient to be performed on any edge-colored complete graph to satisfy a given hereditary property is investigated. The theory for computing the edit distance is extended using random structures and so-called types or colored homomorphisms of graphs.

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arXiv2016

On weighted Ramsey numbers

The weighted Ramsey number, ${\rm wR}(n,k)$, is the minimum $q$ such that there is an assignment of nonnegative real numbers (weights) to the edges of $K_n$ with the total sum of the weights equal to ${n\choose 2}$ and there is a Red/Blue coloring of edges of the same $K_n$, such that in any complete $k$-vertex subgraph $H$, of $K_n$, the sum of the weights on Red edges in $H$ is at most $q$ and the sum of the weights on Blue edges in $H$ is at most $q$. This concept was introduced recently by Fujisawa and Ota. We provide new bounds on ${\rm wR}(n,k)$, for $k\geq 4$ and $n$ large enough and show that determining ${\rm wR}(n,3)$ is asymptotically equivalent to the problem of finding the fractional packing number of monochromatic triangles in colorings of edges of complete graphs with two colors.

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arXiv2016

Avoiding rainbow induced subgraphs in vertex-colorings

For a fixed graph $H$ on $k$ vertices, and a graph $G$ on at least $k$ vertices, we write $G\rightarrow H$ if in any vertex-coloring of $G$ with $k$ colors, there is an induced subgraph isomorphic to $H$ whose vertices have distinct colors. In other words, if $G\rightarrow H$ then a totally multicolored induced copy of $H$ is unavoidable in any vertex-coloring of $G$ with $k$ colors. In this paper, we show that, with a few notable exceptions, for any graph $H$ on $k$ vertices and for any graph $G$ which is not isomorphic to $H$, $G\not\!\rightarrow H$. We explicitly describe all exceptional cases. This determines the induced vertex-anti-Ramsey number for all graphs and shows that totally multicolored induced subgraphs are, in most cases, easily avoidable.

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arXiv2016

Chromatic number of ordered graphs with forbidden ordered subgraphs

It is well-known that the graphs not containing a given graph H as a subgraph have bounded chromatic number if and only if H is acyclic. Here we consider ordered graphs, i.e., graphs with a linear ordering on their vertex set, and the function f(H) = sup{chi(G) | G in Forb(H)} where Forb(H) denotes the set of all ordered graphs that do not contain a copy of H. If H contains a cycle, then as in the case of unordered graphs, f(H) is infinity. However, in contrast to the unordered graphs, we describe an infinite family of ordered forests H with infinite f(H). An ordered graph is crossing if there are two edges uv and u'v' with u < u' < v < v'. For connected crossing ordered graphs H we reduce the problem of determining whether f(H) is finite to a family of so-called monotonically alternating trees. For non-crossing H we prove that f(H) is finite if and only if H is acyclic and does not contain a copy of any of the five special ordered forests on four or five vertices, which we call bonnets. For such forests H, we show that f(H) <= 2^|V(H)| and that f(H) <= 2|V(H)|-3 if H is connected.

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arXiv2015

Boolean lattices: Ramsey properties and embeddings

A subposet $Q'$ of a poset $Q$ is a copy of a poset $P$ if there is a bijection $f$ between elements of $P$ and $Q'$ such that $x\leq y$ in $P$ iff $f(x)\leq f(y)$ in $Q'$. For posets $P, P'$, let the poset Ramsey number $R(P,P')$ be the smallest $N$ such that no matter how the elements of the Boolean lattice $Q_N$ are colored red and blue, there is a copy of $P$ with all red elements or a copy of $P'$ with all blue elements. We provide some general bounds on $R(P,P')$ and focus on the situation when $P$ and $P'$ are both Boolean lattices. In addition, we give asymptotically tight bounds for the number of copies of $Q_n$ in $Q_N$ and for a multicolor version of a poset Ramsey number.

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arXiv2015

Density of Range Capturing Hypergraphs

For a finite set $X$ of points in the plane, a set $S$ in the plane, and a positive integer $k$, we say that a $k$-element subset $Y$ of $X$ is captured by $S$ if there is a homothetic copy $S'$ of $S$ such that $X\cap S' = Y$, i.e., $S'$ contains exactly $k$ elements from $X$. A $k$-uniform $S$-capturing hypergraph $H = H(X,S,k)$ has a vertex set $X$ and a hyperedge set consisting of all $k$-element subsets of $X$ captured by $S$. In case when $k=2$ and $S$ is convex these graphs are planar graphs, known as convex distance function Delaunay graphs. In this paper we prove that for any $k\geq 2$, any $X$, and any convex compact set $S$, the number of hyperedges in $H(X,S,k)$ is at most $(2k-1)|X| - k^2 + 1 - \sum_{i=1}^{k-1}a_i$, where $a_i$ is the number of $i$-element subsets of $X$ that can be separated from the rest of $X$ with a straight line. In particular, this bound is independent of $S$ and indeed the bound is tight for all "round" sets $S$ and point sets $X$ in general position with respect to $S$. This refines a general result of Buzaglo, Pinchasi and Rote stating that every pseudodisc topological hypergraph with vertex set $X$ has $O(k^2|X|)$ hyperedges of size $k$ or less.

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arXiv2015

Splitting Planar Graphs of Girth 6 into Two Linear Forests with Short Paths

Recently, Borodin, Kostochka, and Yancey (On $1$-improper $2$-coloring of sparse graphs. Discrete Mathematics, 313(22), 2013) showed that the vertices of each planar graph of girth at least $7$ can be $2$-colored so that each color class induces a subgraph of a matching. We prove that any planar graph of girth at least $6$ admits a vertex coloring in $2$ colors such that each monochromatic component is a path of length at most $14$. Moreover, we show a list version of this result. On the other hand, for each positive integer $t\geq 3$, we construct a planar graph of girth $4$ such that in any coloring of vertices in $2$ colors there is a monochromatic path of length at least $t$. It remains open whether each planar graph of girth $5$ admits a $2$-coloring with no long monochromatic paths.

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arXiv2015

Conditions on Ramsey non-equivalence

Given a graph H, a graph G is called a Ramsey graph of H if there is a monochromatic copy of H in every coloring of the edges of G with two colors. Two graphs G, H are called Ramsey equivalent if they have the same set of Ramsey graphs. Fox et al. [J. Combin. Theory Ser. B 109 (2014), 120--133] asked whether there are two non-isomorphic connected graphs that are Ramsey equivalent. They proved that a clique is not Ramsey equivalent to any other connected graph. Results of Nesetril et al. showed that any two graphs with different clique number [Combinatorica 1(2) (1981), 199--202] or different odd girth [Comment. Math. Univ. Carolin. 20(3) (1979), 565--582] are not Ramsey equivalent. These are the only structural graph parameters we know that "distinguish" two graphs in the above sense. This paper provides further supportive evidence for a negative answer to the question of Fox et al. by claiming that for wide classes of graphs, chromatic number is a distinguishing parameter. In addition, it is shown here that all stars and paths and all connected graphs on at most 5 vertices are not Ramsey equivalent to any other connected graph. Moreover two connected graphs are not Ramsey equivalent if they belong to a special class of trees or to classes of graphs with clique-reduction properties.

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arXiv2014

Spectrum of mixed bi-uniform hypergraphs

A mixed hypergraph is a triple $H=(V,\mathcal{C},\mathcal{D})$, where $V$ is a set of vertices, $\mathcal{C}$ and $\mathcal{D}$ are sets of hyperedges. A vertex-coloring of $H$ is proper if $C$-edges are not totally multicolored and $D$-edges are not monochromatic. The feasible set $S(H)$ of $H$ is the set of all integers, $s$, such that $H$ has a proper coloring with $s$ colors. Bujtás and Tuza [Graphs and Combinatorics 24 (2008), 1--12] gave a characterization of feasible sets for mixed hypergraphs with all $C$- and $D$-edges of the same size $r$, $r\geq 3$. In this note, we give a short proof of a complete characterization of all possible feasible sets for mixed hypergraphs with all $C$-edges of size $\ell$ and all $D$-edges of size $m$, where $\ell, m \geq 2$. Moreover, we show that for every sequence $(r(s))_{s=\ell}^n$, $n \geq \ell$, of natural numbers there exists such a hypergraph with exactly $r(s)$ proper colorings using $s$ colors, $s = \ell,\ldots,n$, and no proper coloring with more than $n$ colors. Choosing $\ell = m=r$ this answers a question of Bujtás and Tuza, and generalizes their result with a shorter proof.

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arXiv2013

Online and size anti-Ramsey numbers

A graph is properly edge-colored if no two adjacent edges have the same color. The smallest number of edges in a graph any of whose proper edge colorings contains a totally multicolored copy of a graph H is the size anti-Ramsey number AR_s(H) of H. This number in offline and online setting is investigated here.

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arXiv2013

Twins in graphs

A basic pigeonhole principle insures an existence of two objects of the same type if the number of objects is larger than the number of types. Can such a principle be extended to a more complex combinatorial structure? Here, we address such a question for graphs. We call two disjoint subsets $A, B$ of vertices \emph{\textbf{twins}} if they have the same cardinality and induce subgraphs of the same size. Let $t(G)$ be the largest $k$ such that $G$ has twins on $k$ vertices each. We provide the bounds on $t(G)$ in terms of the number of edges and vertices using discrepancy results for induced subgraphs. In addition, we give conditions under which $t(G)= |V(G)|/2$ and show that if $G$ is a forest then $t(G) \geq |V(G)|/2 - 1$.

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arXiv2013

Multicolor Ramsey numbers for triple systems

Given an $r$-uniform hypergraph $H$, the multicolor Ramsey number $r_k(H)$ is the minimum $n$ such that every $k$-coloring of the edges of the complete $r$-uniform hypergraph $K_n^r$ yields a monochromatic copy of $H$. We investigate $r_k(H)$ when $k$ grows and $H$ is fixed. For nontrivial 3-uniform hypergraphs $H$, the function $r_k(H)$ ranges from $\sqrt{6k}(1+o(1))$ to double exponential in $k$. We observe that $r_k(H)$ is polynomial in $k$ when $H$ is $r$-partite and at least single-exponential in $k$ otherwise. Erdős, Hajnal and Rado gave bounds for large cliques $K_s^r$ with $s\ge s_0(r)$, showing its correct exponential tower growth. We give a proof for cliques of all sizes, $s>r$, using a slight modification of the celebrated stepping-up lemma of Erdős and Hajnal. For 3-uniform hypergraphs, we give an infinite family with sub-double-exponential upper bound and show connections between graph and hypergraph Ramsey numbers. Specifically, we prove that $$r_k(K_3)\le r_{4k}(K_4^3-e)\le r_{4k}(K_3)+1,$$ where $K_4^3-e$ is obtained from $K_4^3$ by deleting an edge. We provide some other bounds, including single-exponential bounds for $F_5=\{abe,abd,cde\}$ as well as asymptotic or exact values of $r_k(H)$ when $H$ is the bow $\{abc,ade\}$, kite $\{abc,abd\}$, tight path $\{abc,bcd,cde\}$ or the windmill $\{abc,bde,cef,bce\}$. We also determine many new "small" Ramsey numbers and show their relations to designs. For example, the lower bound for $r_6(kite)=8$ is demonstrated by decomposing the triples of $[7]$ into six partial STS (two of them are Fano planes).

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arXiv2012

A regularity lemma and twins in words

For a word $S$, let $f(S)$ be the largest integer $m$ such that there are two disjoints identical (scattered) subwords of length $m$. Let $f(n, Σ) = \min \{f(S): S \text{is of length} n, \text{over alphabet} Σ\}$. Here, it is shown that \[2f(n, \{0,1\}) = n-o(n)\] using the regularity lemma for words. I.e., any binary word of length $n$ can be split into two identical subwords (referred to as twins) and, perhaps, a remaining subword of length $o(n)$. A similar result is proven for $k$ identical subwords of a word over an alphabet with at most $k$ letters.

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