arXiv subjects

Yair Caro

Publications and source records attributed to Yair Caro.

At least 37 records · Page 2Linked to original sources

arXiv2024

Monochromatic graph decompositions inspired by anti-Ramsey theory and the odd-coloring problem

We consider extremal edge-coloring problems inspired by the theory of anti-Ramsey / rainbow coloring, and further by odd-colorings and conflict-free colorings. Let $G$ be a graph, and $F$ any given family of graphs. For every integer $n \geq |G|$, let $f(n,G|F)$ denote the smallest integer $k$ such that any edge coloring of the complete graph $K_n$ with at least $k$ colors forces a copy of $G$ in which each color class induces a member of $F$. Observe that in anti-Ramsey problems each color class is a single edge; i.e., $F=\{K_2\}$. In our previous paper [arXiv:2405.19812], attention was given mostly to the case where $F$ is hereditary under subgraph inclusion. In the present work we consider coloring problems inspired by odd-coloring and conflict-free coloring. As we shall see, dealing with these problems requires distinct additional tools to those used in our first paper on the subject. Among the many results introduced in this paper, we mention: (1) For every graph $G$, there exists a constant $c=c(G)$ such that in any edge coloring of $K_n$ with at least $cn$ colors there is a copy of $G$ in which every vertex $v$ is incident with an edge whose color appears only once among all edges incident with $v$. (2) In sharp contrast to the above result we prove that if $F$ is the class of all odd graphs (having vertices with odd degrees only) then $f(n,K_k|F)=(1+o(1))$ex$(n,K_{\lceil k/2 \rceil})$, which is quadratic for $k \geq 5$. (3) We exactly determine $f(n,G|F)$ for small graphs when $F$ belongs to several families representing various odd/even coloring constraints.

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arXiv2024

Monochromatic graph decompositions inspired by anti-Ramsey colorings

We consider coloring problems inspired by the theory of anti-Ramsey / rainbow colorings that we generalize to a far extent. Let $\mathcal{F}$ be a hereditary family of graphs; i.e., if $H\in \mathcal{F}$ and $H'\subset H$ then also $H'\subset \mathcal{F}$. For a graph $G$ and any integer $n \geq |G|$, let $f(n,G|\mathcal{F})$ denote the smallest number $k$ of colors such that any edge coloring of $K_n$ with at least $k$ colors forces a copy of $G$ in which each color class induces a member of $\mathcal{F}$. The case $\mathcal{F} = \{K_2\}$ is the notorious anti-Ramsey / rainbow coloring problem introduced by Erdős, Simonovits and Sós in 1973. Using the $\mathcal{F}$-deck of $G$, $D(G|\mathcal{F}) = \{ H : H = G - D, \, D \in \mathcal{F}\}$, we define $χ_\mathcal{F}(G) = \min \{ χ(H) : H \in D(G|\mathcal{F}) \}$. The main theorem we prove is: Suppose $\mathcal{F}$ is a hereditary family of graphs, and let $G$ be a graph not a member of $\mathcal{F}$. (1) If $χ_\mathcal{F}(G) \geq 3$, then $f(n, G |\mathcal{F}) = (1+o(1)) \, ex(n, K_{χ_\mathcal{F}(G)})$. (2) Otherwise $f(n, G |\mathcal{F}) = o(n^2)$. Among the families covered by this theorem are: matchings, acyclic graphs, planar and outerplanar graphs, $d$-degenerate graphs, graphs with chromatic number at most $k$, graphs with bounded maximum degree, and many more. We supply many concrete examples to demonstrate the wide range of applications of the main theorem; the next result is a representative of these examples. For $p \geq 5$ and $\mathcal{F} = \{ tK_2 : t \geq 1 \}$, we have $f(n,K_p |\mathcal{F}) = (1+o(1)) \, ex(n, K_{\lceil p/2 \rceil})$; this means a properly colored copy of $K_p$. In other words, a certain number of colors forces nearly twice as large properly edge-colored complete subgraphs as rainbow ones.

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arXiv2024

On $(r,c)$-constant, planar and circulant graphs

This paper concerns $(r,c)$-constant graphs, which are $r$-regular graphs in which the subgraph induced by the open neighbourhood of every vertex has precisely $c$ edges. The family of $(r,c)$-graphs contains vertex-transitive graphs (and in particular Cayley graphs), graphs with constant link (sometimes called locally isomorphic graphs), $(r,b)$-regular graphs, strongly regular graphs, and much more. This family was recently introduced in [arXiv:2312.08777] serving as important tool in constructing flip graphs [arXiv:2312.08777, arXiv:2401.02315]. In this paper we shall mainly deal with the following: i. Existence and non-existence of $(r, c)$-planar graphs. We completely determine the cases of existence and non-existence of such graphs and supply the smallest order in the case when they exist. ii. We consider the existence of $(r, c)$-circulant graphs. We prove that for $c \equiv 2 \ (\mathrm{mod} \ 3)$ no $(r,c)$-circulant graph exists and that for $c \equiv 0, 1 \ (\mathrm{mod} \ 3)$, $c > 0$ and $r \geq 6 + \sqrt{\frac{8c - 5}{3}}$ there exists $(r,c)$-circulant graphs. Moreover for $c = 0$ and $r \geq 1$, $(r, 0)$-circulants exist. iii. We consider the existence and non-existence of small $(r,c)$-constant graphs, supplying a complete table of the smallest order of graphs we found for $0 \leq c \leq \binom{r}{2}$ and $r \leq 6$. We shall also determine all the cases in this range for which $(r,c)$-constant graphs don't exist. We establish a public database of $(r,c)$-constant graphs for varying $r$, $c$ and order.

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arXiv2024

Edge mappings of graphs: Turán type parameters

In this paper, we address problems related to parameters concerning edge mappings of graphs. The quantity $h(n,G)$ is defined to be the maximum number of edges in an $n$-vertex graph $H$ such that there exists a mapping $f: E(H)\rightarrow E(H)$ with $f(e)\neq e$ for all $e\in E$ and further in all copies $G'$ of $G$ in $H$ there exists $e\in E(G')$ with $f(e)\in E(G')$. Among other results, we determine $h(n, G)$ when $G$ is a matching and $n$ is large enough. As a related concept, we say that $H$ is unavoidable for $G$ if for any mapping $f: E(H)\rightarrow E(H)$ with $f(e)\neq e$ there exists a copy $G'$ of $G$ in $H$ such that $f(e)\notin E(G')$ for all $e\in E(G)$. The set of minimal unavoidable graphs for $G$ is denoted by $\mathcal{M}(G)$. We prove that if $F$ is a forest, then $\mathcal{M}(F)$ is finite if and only if $F$ is a matching, and we conjecture that for all non-forest graphs $G$, the set $\mathcal{M}(G)$ is infinite. Several other parameters are defined with basic results proved. Lots of open problems remain.

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arXiv2023

Flip colouring of graphs

It is proved that for integers $b, r$ such that $3 \leq b < r \leq \binom{b+1}{2} - 1$, there exists a red/blue edge-colored graph such that the red degree of every vertex is $r$, the blue degree of every vertex is $b$, yet in the closed neighborhood of every vertex there are more blue edges than red edges. The upper bound $r \le \binom{b+1}{2}-1$ is best possible for any $b \ge 3$. We further extend this theorem to more than two colours, and to larger neighbourhoods. A useful result required in some of our proofs, of independent interest, is that for integers $r,t$ such that $0 \leq t \le \frac{r^2}{2} - 5r^{3/2}$, there exists an $r$-regular graph in which each open neighborhood induces precisely $t$ edges. Several explicit constructions are introduced and relationships with constant linked graphs, $(r,b)$-regular graphs and vertex transitive graphs are revealed.

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arXiv2023

The Feasibility Problem -- the family ${\cal F}$$(G)$ of all induced $G$-free graphs

An infinite family of graphs ${\cal F}$ is called feasible if for any pair of integers $(n,m)$, $n \geq 1$, $0 \leq m \leq \binom{n}{2}$, there is a member $G \in {\cal F}$ such that $G$ has $n$ vertices and $m$ edges. We prove that given a graph $G$, the family ${\cal F}$$(G)$ of all induced $G$-free graphs is feasible if and only if $G$ is not $K_k$, $K_k\backslash K_2$, $\overline{K_k}$, $\overline{K_k\backslash K_2}$, for $k \geq 2$.

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arXiv2023

Counting Connected Partitions of Graphs

Motivated by the theorem of Gy\H ori and Lovász, we consider the following problem. For a connected graph $G$ on $n$ vertices and $m$ edges determine the number $P(G,k)$ of unordered solutions of positive integers $\sum_{i=1}^k m_i = m$ such that every $m_i$ is realized by a connected subgraph $H_i$ of $G$ with $m_i$ edges such that $\cup_{i=1}^kE(H_i)=E(G)$. We also consider the vertex-partition analogue. We prove various lower bounds on $P(G,k)$ as a function of the number $n$ of vertices in $G$, as a function of the average degree $d$ of $G$, and also as the size $\mathrm{CMC}_r(G)$ of $r$-partite connected maximum cuts of $G$. Those three lower bounds are tight up to a multiplicative constant. We also prove that the number $π(G,k)$ of unordered $k$-tuples with $\sum_{i=1}^kn_i=n$, that are realizable by vertex partitions into $k$ connected parts of respective sizes $n_1,n_2,\dots,n_k$, is $Ω(d^{k-1})$.

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arXiv2023

The evolution of unavoidable bi-chromatic patterns and extremal cases of balanceability

We study the color patterns that, for $n$ sufficiently large, are unavoidable in $2$-colorings of the edges of a complete graph $K_n$ with respect to $\min \{e(R), e(B)\}$, where $e(R)$ and $e(B)$ are the numbers of red and, respectively, blue edges. More precisely, we determine how such unavoidable patterns evolve from the case without restriction in the coloring, namely that $\min \{e(R), e(B)\} \ge 0$ (given by Ramsey's theorem), to the highest possible restriction, namely that $|e(R) - e(B)| \le 1$. We also investigate the effect of forbidding certain sub-structures in each color. In particular, we show that, in $2$-colorings whose graphs induced by each of the colors are both free from an induced matching on $r$ edges, the appearance of the unavoidable patterns is already granted with a much weaker restriction on $\min \{e(R), e(B)\}$. We finish analyzing the consequences of these results to the balancing number $bal(n,G)$ of a graph $G$ (i.e. the minimum $k$ such that every $2$-edge coloring of $K_n$ with $\min \{e(R), e(B)\} > k$ contains a copy of $G$ with half the edges in each color), and show that, for every $\varepsilon > 0$, there are graphs $G$ with $bal(n,G) \ge c n^{2-\varepsilon}$, which is the highest order of magnitude that is possible to achieve, as well as graphs where $bal(n,G) \le c(G)$, where $c(G)$ is a constant that depends only $G$. We characterize the latter ones.

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arXiv2023

Graphs isomorphisms under edge-replacements and the family of amoebas

This paper offers a systematic study of a family of graphs called amoebas. Amoebas recently emerged from the study of forced patterns in $2$-colorings of the edges of the complete graph in the context of Ramsey-Turan theory and played an important role in extremal zero-sum problems. Amoebas are graphs with a unique behavior with regards to the following operation: Let $G$ be a graph and let $e\in E(G)$ and $e'\in E(\overline{G})$. If the graph $G'=G-e+e'$ is isomorphic to $G$, we say $G'$ is obtained from $G$ by performing a \emph{feasible edge-replacement}. We call $G$ a \emph{local amoeba} if, for any two copies $G_1$, $G_2$ of $G$ on the same vertex set, $G_1$ can be transformed into $G_2$ by a chain of feasible edge-replacements. On the other hand, $G$ is called \emph{global amoeba} if there is an integer $t_0 \ge 0$ such that $G \cup tK_1$ is a local amoeba for all $t \ge t_0$. To model the dynamics of the feasible edge-replacements of $G$, we define a group ${\rm Fer}(G)$ that satisfies that $G$ is a local amoeba if and only if ${\rm Fer}(G) \cong S_n$, where $n$ is the order of $G$. Via this algebraic setting, a deeper understanding of the structure of amoebas and their intrinsic properties comes into light. Moreover, we present different constructions that prove the richness of these graph families showing, among other things, that any connected graph can be a connected component of a global amoeba, that global amoebas can be very dense and that they can have, in proportion to their order, large clique and chromatic numbers. Also, a family of global amoeba trees with a Fibonacci-like structure and with arbitrary large maximum degree is constructed.

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arXiv2022

Connected Turán number of trees

As a variant of the much studied Turán number, $ex(n,F)$, the largest number of edges that an $n$-vertex $F$-free graph may contain, we introduce the connected Turán number $ex_c(n,F)$, the largest number of edges that an $n$-vertex connected $F$-free graph may contain. We focus on the case where the forbidden graph is a tree. The celebrated conjecture of Erdős and Sós states that for any tree $T$, we have $ex(n,T)\le(|T|-2)\frac{n}{2}$. We address the problem how much smaller $ex_c(n,T)$ can be, what is the smallest possible ratio of $ex_c(n,T)$ and $(|T|-2)\frac{n}{2}$ as $|T|$ grows. We also determine the exact value of $ex_c(n,T)$ for small trees, in particular for all trees with at most six vertices. We introduce general constructions of connected $T$-free graphs based on graph parameters as longest path, matching number, branching number, etc.

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arXiv2022

Higher Degree Erdos-Ginzburg-Ziv Constants

We generalize the notion of Erdős-Ginzburg-Ziv constants -- along the same lines we generalized in earlier work the notion of Davenport constants -- to a ``higher degree" and obtain various lower and upper bounds. These bounds are sometimes exact as is the case for certain finite commutative rings of prime power cardinality. We also consider to what extent a theorem due independently to W.D.~Gao and the first author that relates these two parameters extends to this higher degree setting. Two simple examples that capture the essence of these higher degree Erdős-Ginzburg-Ziv constants are the following. 1) Let $ν_p(m)$ denote the $p-$adic valuation of the integer $m$. Suppose we have integers $t | {m \choose 2}$ and $n=t+2^{ν_2(m)}$, then every sequence $S$ over ${\mathbb Z}_2$ of length $|S| \geq n$ contains a subsequence $S'$ of length $t$ for which $\sum_{a_{i_1},\ldots, a_{i_m} \in S'} a_{i_1}\cdots a_{i_m} \equiv 0 \pmod{2}$, and this is sharp. 2) Suppose $k=3^α$ for some integer $α\geq 2$. Then every sequence $S$ over ${\mathbb Z}_3$ of length $|S| \geq k+6$ contains a subsequence $S'$ of length $k$ for which $\sum_{a_h, a_i, a_j \in S'} a_ha_ia_j \equiv 0 \pmod{3}$. These examples illustrate that if a sequence of elements from a finite commutative ring is long enough, certain symmetric expressions (symmetric polynomials) have to vanish on the elements of a subsequence of prescribed length. The Erdős-Ginzburg-Ziv Theorem is just the case where a sequence of length $2n-1$ over ${\mathbb Z}_n$ contains a subsequence $S'=(a_1, \ldots, a_n)$ of length $n$ that vanishes when substituted in the linear symmetric polynomial $a_1+\cdots+a_n.$

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arXiv2022

Remarks on odd colorings of graphs

A proper vertex coloring $φ$ of graph $G$ is said to be odd if for each non-isolated vertex $x\in V(G)$ there exists a color $c$ such that $φ^{-1}(c)\cap N(x)$ is odd-sized. The minimum number of colors in any odd coloring of $G$, denoted $χ_o(G)$, is the odd chromatic number. Odd colorings were recently introduced in [M.~Petruševski, R.~Škrekovski: \textit{Colorings with neighborhood parity condition}]. Here we discuss various basic properties of this new graph parameter, characterize acyclic graphs and hypercubes in terms of odd chromatic number, establish several upper bounds in regard to degenericity or maximum degree, and pose several questions and problems.

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arXiv2022

The feasibility problem for line graphs

We consider the following feasibility problem: given an integer $n \geq 1$ and an integer $m$ such that $0 \leq m \leq \binom{n}{2}$, does there exist a line graph $L = L(G)$ with exactly $n$ vertices and $m$ edges ? We say that a pair $(n,m)$ is non-feasible if there exists no line graph $L(G)$ on $n$ vertices and $m$ edges, otherwise we say $(n,m)$ is a feasible pair. Our main result shows that for fixed $n\geq 5$, the values of $m$ for which $(n, m)$ is a non-feasible pair, form disjoint blocks of consecutive integers which we completely determine. On the other hand we prove, among other things, that for the more general family of claw-free graphs (with no induced $K_{1,3}$-free subgraph), all $(n,m)$-pairs in the range $0 \leq m \leq \binom{n}{2}$ are feasible pairs.

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arXiv2022

Remarks on proper conflict-free colorings of graphs

A vertex coloring of a graph is said to be \textit{conflict-free} with respect to neighborhoods if for every non-isolated vertex there is a color appearing exactly once in its (open) neighborhood. As defined in [Fabrici et al., \textit{Proper Conflict-free and Unique-maximum Colorings of Planar Graphs with Respect to Neighborhoods}, arXiv preprint], the minimum number of colors in any such proper coloring of graph $G$ is the PCF chromatic number of $G$, denoted $χ_{\mathrm{pcf}}(G)$. In this paper, we determine the value of this graph parameter for several basic graph classes including trees, cycles, hypercubes and subdivisions of complete graphs. We also give upper bounds on $χ_{\mathrm{pcf}}(G)$ in terms of other graph parameters. In particular, we show that $χ_{\mathrm{pcf}}(G) \leq5Δ(G)/2$ and characterize equality. Several sufficient conditions for PCF $k$-colorability of graphs are established for $4\le k\le 6$. The paper concludes with few open problems.

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arXiv2022

Higher Degree Davenport Constants over Finite Commutative Rings

We generalize the notion of Davenport constants to a `higher degree' and obtain various lower and upper bounds, which are sometimes exact as is the case for certain finite commutative rings of prime power cardinality. Two simple examples that capture the essence of these higher degree Davenport constants are the following. 1) Suppose $n = 2^k$, then every sequence of integers $S$ of length $2n$ contains a subsequence $S'$ of length at least two such that $\sum_{a_i,a_j \in S'} a_ia_j \equiv 0 \pmod{n}$ and the bound is sharp. 2) Suppose $n \equiv1 \pmod{2}$, then every sequence of integers $S$ of length $2n -1$ contains a subsequence $S'$ of length at least two such that $\sum_{a_i,a_j \in S'} a_ia_j \equiv 0 \pmod{n}$. These examples illustrate that if a sequence of elements from a finite commutative ring is long enough, certain symmetric expressions have to vanish on the elements of a subsequence.

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arXiv2022

Index of Parameters of Iterated Line Graphs

Let $G$ be a prolific graph, that is a finite connected simple graph which is not isomorphic to a cycle nor a path nor the star graph $K_{1,3}$. The line-graph of $G$, denoted by $L(G)$, is defined by having its vertex-set equal to the edge-set of $G$ and two vertices of $L(G)$ are adjacent if the corresponding edges are adjacent in $G$. For a positive integer $k$, the iterated line-graph $L^k(G)$ is defined recursively by $L^k(G)=L(L^{k-1}(G))$. We consider fifteen graph parameters and study their behaviour when the operation of taking the line-graph is iterated. We shall first show that all of these parameters are unbounded. This idea is motivated by a well-known result of van Rooij and Wilf that says that the number of vertices is unbounded if and only if the graph is prolific. We then study of the value of $k(P,{\mathcal F})$, which is the index of a family of prolific graphs with regards to a given graph parameter $P(G)$. For a given parameter $P(G)$, the index of $G$ is denoted by $ind(P,G) = \min \{ r : P(G) < P(L^r(G) \}$. Now for a family $\mathcal F$ of prolific graphs, the index of the family is $k(P,\mathcal{F}) = \max \{ ind(P,G) : G \in \mathcal F\}$. The problem of determining the index of a parameter over the family of prolific graphs is motivated by a result of Chartrand who showed that it could require $k=|V(G)|-3$ iterations for $L^k(G)$ to have a hamiltonian cycle. For twelve of the parameters considered, we exactly determine $k(P,\mathcal F)$ where $\mathcal F$ is the family of all prolific graphs, and for some parameters we also characterize the class of prolific graphs realizing the extremal value $k(P,\mathcal F$). Interesting open problems remain, in particular completing the determination of $k(P,\mathcal F)$ for the three parameters: the independence number, independent domination number and domination number.

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arXiv2021

Recursive constructions of amoebas

Global amoebas are a wide and rich family of graphs that emerged from the study of certain Ramsey-Turán problems in $2$-colorings of the edges of the complete graph $K_n$ that deal with the appearance of unavoidable patterns once a certain amount of edges in each color is guaranteed. Indeed, it turns out that, as soon as such coloring constraints are satisfied and if $n$ is sufficiently large, then every global amoeba can be found embedded in $K_n$ such that it has half its edges in each color. Even more surprising, every bipartite global amoeba $G$ is unavoidable in every tonal-variation, meaning that, for any pair of integers $r, b$ such that $r + b $ is the number of edges of $G$, there is a subgraph of $K_n$ isomorphic to $G$ with $r$ edges in the first color and $b$ edges in the second. The feature that makes global amoebas work are one-by-one edge replacements that leave the structure of the graph invariant. By means of a group theoretical approach, the dynamics of this feature can be modeled. As a counterpart to the global amoebas that "live" inside a possibly large complete graph $K_n$, we also consider local amoebas which are spanning subgraphs of $K_n$ with the same feature. In an effort to highlight their richness and versatility, we present here three different recursive constructions of amoebas, two of them yielding interesting families per se and one of them offering a wide range of possibilities.

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arXiv2021

Conjecture of TxGraffiti: Independence, domination, and matchings

TxGraffiti is an automated conjecturing program that produces graph theoretic conjectures in the form of conjectured inequalities. This program written and maintained by the second author since 2017 was inspired by the successes of previous automated conjecturing programs including Fajtlowicz's GRAFFITI and DeLaViña's GRAFFITI.pc. In this paper we prove and generalize several conjectures generated by TxGraffiti when it was prompted to conjecture on the \emph{independence number}, the \emph{domination number}, and the \emph{matching number} (and generalizations of each of these graph invariants). Moreover, in several instances we also show the proposed inequalities relating these graph invariants are sharp.

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