Oleg Pikhurko
Caro and Yuster (Electronic J.Comb 7 (2000)) studied a generalization of the Turan problem, where a certain function (instead of the size) of an F-free graph of order n has to be maximized. We prove that for a wide class of functions the asymptotics of the maximum is given by complete partite graphs.
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We prove a conjecture of Boros, Caro, Füredi and Yuster on the maximum number of edges in a 2-connected graph without repeated cycle lengths, which is a restricted version of a longstanding problem of Erdős. Our proof together with the matched lower bound construction of Boros, Caro, Füredi and Yuster show that this problem can be conceptually reduced to the seminal problem of finding the maximum Sidon sequences in number theory.
Yuster and Caro initiated the study of the sum of powers of the degrees of graphs with forbidden subgraphs. We settle two of their conjectures.
Caro, West, and Yuster studied how $r$-uniform hypergraphs can be oriented in such a way that (generalizations of) indegree and outdegree are as close to each other as can be hoped. They conjectured an existence result of such orientations for sparse hypergraphs, of which we present a proof.
We prove a conjecture of Yuster and Caro about the sum of the p-powers of the degrees of a graph of order n without a specified even cycle. Our proof is based on a new sufficient condition for long paths, that may be useful in other applications as well.
The inducibility of Turán graphs
本研究证明了Turán图的诱导性由特定Turán图唯一达到，并解决了相关极值图论问题。
Given a positive integer $p$ and a graph $G$ with degree sequence $d_1,\dots,d_n$, we define $e_p(G)=\sum_{i=1}^n d_i^p$. Caro and Yuster introduced a Turán-type problem for $e_p(G)$: Given a positive integer $p$ and a graph $H$, determine the function $ex_p(n,H)$, which is the maximum value of $e_p(G)$ taken over all graphs $G$ on $n$ vertices that do not contain $H$ as a subgraph. Clearly, $ex_1(n,H)=2ex(n,H)$, where $ex(n,H)$ denotes the classical Turán number. Caro and Yuster determined the function $ex_p(n, P_\ell)$ for sufficiently large $n$, where $p\geq 2$ and $P_\ell$ denotes the path…
Answering a question posed by Caro, Hansberg, Lauri, and Zarb, we show that for every positive integer $n$ and every function $σ\colon E(K_{4n})\to\{-1,1\}$ with $σ\left(E(K_{4n})\right)=0$, there is a perfect matching $M$ in $K_{4n}$ with $σ(M)=0$. Strengthening a result of Caro and Yuster, we show that for every positive integer $n$ and every function $σ\colon E(K_{4n})\to\{-1,1\}$ with $\left|σ\left(E(K_{4n})\right)\right|<n^2+11n+2,$ there is a perfect matching $M$ in $K_{4n}$ with $|σ(M)|\leq 2$. Both these results are best possible.
The concept of monochromatic connection of graphs was introduced by Caro and Yuster in 2011. Recently, a lot of results have been published about it. In this survey, we attempt to bring together all the results that dealt with it. We begin with an introduction, and then classify the results into the following categories: monochromatic connection coloring of edge-version, monochromatic connection coloring of vertex-version, monochromatic index, monochromatic connection coloring of total-version.
Improving the Caro-Wei bound and applications to Turán stability
本文改进了Caro-Wei界，证明了基于团中单纯顶点比例的独立集大小新下界，并据此推导出新的Turán稳定性结果，为边数接近极值的$K_{r+1}$-free图是$r$-部图提供了新证明。
The Connected Bipartite Turán Problem for Long Cycles and Paths
本文针对Caro等人提出的连通二部图极值边数问题，精确确定了不含特定长度路径的连通二部图的极值边数与结构，还建立了不含长环的2-连通二部图的精确结构，并应用结果简洁证明了经典定理。
Given $p\geq 0$ and a graph $G$ whose degree sequence is $d_1,d_2,\ldots,d_n$, let $e_p(G)=\sum_{i=1}^n d_i^p$. Caro and Yuster introduced a Turán-type problem for $e_p(G)$: given $p\geq 0$, how large can $e_p(G)$ be if $G$ has no subgraph of a particular type. Denote by $ex_p(n,H)$ the maximum value of $e_p(G)$ taken over all graphs with $n$ vertices that do not contain $H$ as a subgraph. Clearly, $ex_1(n,H)=2ex(n,H)$, where $ex(n,H)$ denotes the classical Turán number, i.e., the maximum number of edges among all $H$-free graphs with $n$ vertices. Pikhurko and Taraz generalize this Turán-type…
Spectral Turán-type problems on sparse spanning graphs
本文研究了稀疏生成图中谱极值图的问题，证明了在足够大的n下，当F无孤立顶点且最大度不超过√n/40时，SPEX(n,F)包含于EX(n,F)。同时给出了无符号拉普拉斯谱半径版本的定理，并给出了特殊图类存在性的紧谱条件。
For an integer $k$ at least $2$, and a graph $G$, let $f_k(G)$ be the minimum cardinality of a set $X$ of vertices of $G$ such that $G-X$ has either $k$ vertices of maximum degree or order less than $k$. Caro and Yuster (Discrete Mathematics 310 (2010) 742-747) conjectured that, for every $k$, there is a constant $c_k$ such that $f_k(G)\leq c_k \sqrt{n(G)}$ for every graph $G$. Verifying a conjecture of Caro, Lauri, and Zarb (arXiv:1704.08472v1), we show the best possible result that, if $t$ is a positive integer, and $F$ is a forest of order at most $\frac{1}{6}\left(t^3+6t^2+17t+12\right)$, …
On degree powers in the degenerate Turán problem
本文针对退化Turán问题，基于前人方法得到$ex_{p}(n, \mathcal{F})$的稳定性结果，证明$n$足够大时极值图含指定完全二分图，推导二分图$F$的已知结果并得到多类$F$的新精确结果。
An edge-coloring of a connected graph $G$ is called a {\it monochromatic connection coloring} (MC-coloring, for short), introduced by Caro and Yuster, if there is a monochromatic path joining any two vertices of the graph $G$. Let $mc(G)$ denote the maximum number of colors used in an MC-coloring of a graph $G$. Note that an MC-coloring does not exist if $G$ is not connected, and in this case we simply let $mc(G)=0$. We use $G(n,p)$ to denote the Erdös-Rényi random graph model, in which each of the $\binom{n}{2}$ pairs of vertices appears as an edge with probability $p$ independently from othe…
Graphs With the Same Edge Count in Each Neighborhood
本文研究每个顶点邻域诱导相同边数的正则图存在性，将不存在性界加强到更优值，并利用近似子群引理刻画阿贝尔群 Cayley 图的可达边数，同时部分解决多类型边图的 flip 着色问题。
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