Lior Gishboliner
We complete the characterization of the digraphs $D$ for which the induced $D$-removal lemma has polynomial bounds, answering a question of Alon and Shapira. We also study the analogous problem for $k$-colored complete graphs. In particular, we prove a removal lemma with polynomial bounds for Gallai colorings.
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A {\em kernel by properly colored paths} of an arc-colored digraph $D$ is a set $S$ of vertices of $D$ such that (i) no two vertices of $S$ are connected by a properly colored directed path in $D$, and (ii) every vertex outside $S$ can reach $S$ by a properly colored directed path in $D$. In this paper, we conjecture that every arc-colored digraph with all cycles properly colored has such a kernel and verify the conjecture for unicyclic digraphs, semi-complete digraphs and bipartite tournaments, respectively. Moreover, weaker conditions for the latter two classes of digraphs are given.
On the singularity and the inverse of 3-colored digraphs
本文研究了3色有向图的非奇异性和逆矩阵特性，探讨了非奇异单环和双环有向图的特征，并分析了其逆矩阵的零对角线条件及可逆性。
Kernels in digraphs with colored vertices
该研究提出顶点着色有向图中核的推广概念——上着色核，并推导得出若干有向图族及特定有向图积具备上着色核的充要条件。
We study $k$-colored kernels in $m$-colored digraphs. An $m$-colored digraph $D$ has $k$-colored kernel if there exists a subset $K$ of its vertices such that (i) from every vertex $v\notin K$ there exists an at most $k$-colored directed path from $v$ to a vertex of $K$ and (ii) for every $u,v\in K$ there does not exist an at most $k$-colored directed path between them. In this paper, we prove that for every integer $k\geq 2$ there exists a $% (k+1)$-colored digraph $D$ without $k$-colored kernel and if every directed cycle of an $m$-colored digraph is monochromatic, then it has a $k$-colored …
An $m$-colored digraph $D$ has $k$-colored kernel if there exists a subset $K $ of its vertices such that for every vertex $v\notin K$ there exists an at most $k$-colored directed path from $v$ to a vertex of $K$ and for every $% u,v\in K$ there does not exist an at most $k$-colored directed path between them. In this paper we prove that an $m$-colored semicomplete $r$-partite digraph $D$ has a $k$-colored kernel provided that $r\geq 3$ and {enumerate} [(i)] $k\geq 4,$ [(ii)] $k=3$ and every $\overrightarrow{C}_{4}$ contained in $D$ is at most 2-colored and, either every $\overrightarrow{C}_{5…
A matrix for counting paths in acyclic colored digraphs
本文提出一种与k-着色无环有向图关联的矩阵，并证明其行列式可枚举图中所有路径，从而建立矩阵行列式与路径计数之间的对应关系。
We demonstrate that certain Johnson-type bounds are asymptotically exact for a variety of classes of codes, namely, constant-composition codes, nonbinary constant-weight codes and multiply constant-weight codes. This was achieved via an interesting application of the theory of decomposition of edge-colored digraphs.
Let $D$ be an arc-colored digraph. The arc number $a(D)$ of $D$ is defined as the number of arcs of $D$. The color number $c(D)$ of $D$ is defined as the number of colors assigned to the arcs of $D$. A rainbow triangle in $D$ is a directed triangle in which every pair of arcs have distinct colors. Let $f(D)$ be the smallest integer such that if $c(D)\geq f(D)$, then $D$ contains a rainbow triangle. In this paper we obtain $f(\overleftrightarrow{K}_{n})$ and $f(T_n)$, where $\overleftrightarrow{K}_{n}$ is a complete digraph of order $n$ and $T_n$ is a strongly connected tournament of order $n$.…
A digraph is {\em $d$-dominating} if every set of at most $d$ vertices has a common out-neighbor. For all integers $d\geq 2$, let $f(d)$ be the smallest integer such that the vertices of every 2-edge-colored (finite or infinite) complete digraph (including loops) can be covered by the vertices of at most $f(d)$ monochromatic $d$-dominating subgraphs. Note that the existence of $f(d)$ is not obvious -- indeed, the question which motivated this paper was simply to determine whether $f(d)$ is bounded, even for $d=2$. We answer this question affirmatively for all $d\geq 2$, proving $4\leq f(2)\le …
Let $H$ be a digraph possibly with loops and $D$ a digraph without loops whose arcs are colored with the vertices of $H$ ($D$ is said to be an $H-$colored digraph). If $W=(x_{0},\ldots,x_{n})$ is an open walk in $D$ and $i\in \{1,\ldots,n-1\}$, we say that there is an obstruction on $x_{i}$ if $(color(x_{i-1},x_{i}),color(x_{i},x_{i+1}))\notin A(H)$. If $S\subseteq V(D)$, we say that $S$ is a $(k,l,H)$-kernel by walks if for every pair of different vertices in $S$, every walk between them has at least $k-1$ obstructions, and for every $x\in V(D)\setminus S$ there exists an $xS$-walk with at mo…
Let $H$ be a digraph possibly with loops and $D$ a digraph without loops with a coloring of its arcs $c:A(D) \rightarrow V(H)$ ($D$ is said to be an $H$-colored digraph). A directed path $W$ in $D$ is said to be an $H$-path if and only if the consecutive colors encountered on $W$ form a directed walk in $H$. A subset $N$ of vertices of $D$ is said to be an $H$-kernel if (1) for every pair of different vertices in $N$ there is no $H$-path between them and (2) for every vertex $u$ in V($D$)$\setminus$$N$ there exists an $H$-path in $D$ from $u$ to $N$. Under this definition an $H$-kernel is a ke…
A Mathon-type construction for digraphs and improved lower bounds for Ramsey numbers
本文提出有向图的 Mathon 型构造，证明其与 k 次幂 Paley 有向图的联系，并据此改进多色有向 Ramsey 数的下界。
Recent investigations in computational biology focus on a family of 2-colored digraphs, called 2-colored best match graphs, which naturally arise from rooted phylogenetic trees. Actually the defining properties of such graphs are unexpectedly unusual in graph theory, and they were established only recently after the discovery of their links to evolutionary relatedness via phylogenetic trees. In this paper several results are obtained on 2-colored best match graphs which well fit in the mainstream of graph theory.
Let $H$ be a digraph possibly with loops, $D$ a digraph without loops, and $ρ: A(D) \rightarrow V(H)$ a coloring of $A(D)$ ($D$ is said to be an $H$-colored digraph). If $W=(x_{0}, \ldots , x_{n})$ is a walk in $D$, and $i \in \{ 0, \ldots , n-1 \}$, we say that there is an obstruction on $x_{i}$ whenever $(ρ(x_{i-1}, x_{i}), ρ(x_{i}, x_{i+1})) \notin A(H)$ (when $x_{0} = x_{n}$ the indices are taken modulo $n$). We denote by $O_{H}(W)$ the set $\{ i \in \{0, \ldots , n-1 \} :$ there is an obstruction on $x_{i} \}$. The $H$-length of $W$, denoted by $l_{H}(W)$, is defined by $|O_{H}(W)|+1$ whe…
\qquad A \emph{coloring} of a digraph $D=(V,E)$ is a coloring of its vertices following the rule: Let $uv$ be an arc in $D$. If the tail $u$ is colored first, then the head $v$ should receive a color different from that of $u$. The \emph{dichromatic number} $χ_d(D)$ of $D$ is the minimum number of colors needed in a coloring of $D$. Besides obtaining many results and bounds for $χ_d(D)$ analogous to that of chromatic number of a graph, we prove $χ_d(D)=1$ if $D$ is acyclic. New notions of sequential colorings of graphs/digraphs are introduced. A characterization of acyclic digraph is obtained …
In the paper Graphical complexity of products of permutation groups, M. Grech, A. Jez, A. Kisielewicz have proved that the direct product of automorphism groups of edge-colored graphs is itself the automorphism groups of an edge-colored graph. In this paper, we study the direct product of two permutation groups such that at least one of them fails to be the automorphism group of an edge-colored graph. We find necessary and sufficient conditions for the direct product to be the automorphism group of an edge-colored graph. The same problem is solved for the edge-colored digraphs.
For an edge-colored graph $G$, the minimum color degree of $G$ means the minimum number of colors on edges which are adjacent to each vertex of $G$. We prove that if $G$ is an edge-colored graph with minimum color degree at least $5$ then $V(G)$ can be partitioned into two parts such that each part induces a subgraph with minimum color degree at least $2$. We show this theorem by proving a much stronger form. Moreover, we point out an important relationship between our theorem and Bermond-Thomassen's conjecture in digraphs.
Recent investigations in computational biology have focused on a family of 2-colored digraphs, called 2-colored best match graphs, which naturally arise from rooted phylogenetic trees. Actually the defining properties of such graphs are unusual, and a natural question is whether they also have properties which well fit in structural graph theory. In this paper, we prove that some underlying oriented bipartite graphs of a 2-colored best match graph are acyclic and we point out that the arising topological ordering can efficiently be used for constructing new families of 2-colored best match gra…
Odd Properly Colored Cycles in Edge-Colored Graphs
本文研究边着色图中奇数正常着色圈的存在性判定，并作为副产品给出检测给定边集中具有偶数或奇数条边的图中完美匹配的方法。
Color-Constrained Arborescences in Edge-Colored Digraphs
本文研究边着色有向图中满足各颜色边数约束的根树形图，利用含 $q-1$ 个未定元的符号矩阵行列式推广 Tutte 矩阵树定理，解决固定 $q$ 时的计数、判定、搜索及最小权问题。
Vertex-partitions of 2-edge-colored graphs
针对2边着色图的顶点划分问题，受有向图多数着色启发，提出$k$-多数划分概念，证明其存在4-多数划分、判定3-多数划分是NP完全问题，并给出存在平衡3-多数划分的图的度条件。
A partial automorphism of a finite graph is an isomorphism between its vertex induced subgraphs. The set of all partial automorphisms of a given finite graph forms an inverse monoid under composition (of partial maps). We describe the algebraic structure of such inverse monoids by the means of the standard tools of inverse semigroup theory, namely Green's relations and some properties of the natural partial order, and give a characterization of inverse monoids which arise as inverse monoids of partial graph automorphisms. We extend our results to digraphs and edge-colored digraphs as well.
In 1982 Víctor Neumann-Lara introduced the dichromatic number of a digraph $D$ as the smallest integer $k$ such that the vertices $V$ of $D$ can be colored with $k$ colors and each color class induces an acyclic digraph. Later a flow theory for the dichromatic number transferring Tutte's theory of nowhere-zero flows (NZ-flows) from classic graph colorings has been developed by Hochstättler. The purpose of this paper is to pursue this analogy by introducing a new definition of algebraic Neumann-Lara-flows (NL-flows) and a closed formula for their polynomial. Furthermore we generalize the Equiva…
Generalized DP-colorings of digraphs
本文将有向图的广义染色、变量退化与DP-染色三种概念统一结合，推广了Brooks、Gallai等经典图染色定理及Harutyunyan和Mohar的有向图对应结果，主要贡献是给出有向图DP-染色与变量退化的综合框架。
Best match graphs (BMGs) are vertex-colored digraphs that naturally arise in mathematical phylogenetics to formalize the notion of evolutionary closest genes w.r.t. an a priori unknown phylogenetic tree. BMGs are explained by unique least resolved trees. We prove that the property of a rooted, leaf-colored tree to be least resolved for some BMG is preserved by the contraction of inner edges. For the special case of two-colored BMGs, this leads to a characterization of the least resolved trees (LRTs) of binary-explainable trees and a simple, polynomial-time algorithm for the minimum cardinality…
We present a unified framework on the limits of constraint satisfaction problems (CSPs) and efficient parameter testing which depends only on array exchangeability and the method of cut decomposition without recourse to the weakly regular partitions. In particular, we formulate and prove a representation theorem for compact colored $r$-uniform directed hypergraph ($r$-graph) limits, and apply this to $r$CSP limits. We investigate the sample complexity of testable $r$-graph parameters, we discuss the generalized ground state energies and demonstrate that they are efficiently testable.
Improved bounds for the triangle case of Aharoni's rainbow generalization of the Caccetta-Häggkvist conjecture
本文改进了Aharoni关于Caccetta-Häggkvist猜想彩虹推广的三角形式界，证明了(1.1077,1/3)、(1,0.3988)及(1.3481,1/4)等参数对是三角的。
Alternating Hamiltonian cycles in $2$-edge-colored multigraphs
本文研究2-边着色多重图中的交替哈密顿圈，通过施加局部闭性条件，给出了2-M-闭多重图存在交替哈密顿圈的充要条件，并构造了反例族。
We consider properties of edge-colored vertex-ordered graphs, i.e., graphs with a totally ordered vertex set and a finite set of possible edge colors. We show that any hereditary property of such graphs is strongly testable, i.e., testable with a constant number of queries. We also explain how the proof can be adapted to show that any hereditary property of $2$-dimensional matrices over a finite alphabet (where row and column order is not ignored) is strongly testable. The first result generalizes the result of Alon and Shapira [FOCS'05, SICOMP'08], who showed that any hereditary graph propert…
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