Heterochromatic tree partition number of a complete multipartite graph

On multipartite posets

A poset $\mathbf{P} = (X,\preceq)$ is {\em $m$-partite} if $X$ has a partition $X = X_1 \cup ... \cup X_m$ such that (1) each $X_i$ forms an antichain in $\mathbf{P}$, and (2) $x\prec y$ implies $x\in X_i$ and $y\in X_j$ where $i<j$. In this article we derive a tight asymptotic upper bound on the order dimension of $m$-partite posets in terms of $m$ and their bipartite sub-posets in a constructive and elementary way.

Monochromatic connected matchings in 2-edge-colored multipartite graphs

A matching $M$ in a graph $G$ is connected if all the edges of $M$ are in the same component of $G$. Following Łuczak,there have been many results using the existence of large connected matchings in cluster graphs with respect to regular partitions of large graphs to show the existence of long paths and other structures in these graphs. We prove exact Ramsey-type bounds on the sizes of monochromatic connected matchings in $2$-edge-colored multipartite graphs. In addition, we prove a stability theorem for such matchings.

Partitioning two-coloured complete multipartite graphs into monochromatic paths and cycles

We show that any complete $k$-partite graph $G$ on $n$ vertices, with $k \ge 3$, whose edges are two-coloured, can be covered with two vertex-disjoint monochromatic paths of distinct colours. We prove this under the necessary assumption that the largest partition class of $G$ contains at most $n/2$ vertices. This extends known results for complete and complete bipartite graphs. Secondly, we show that in the same situation, all but $o(n)$ vertices of the graph can be covered with two vertex-disjoint monochromatic cycles of distinct colours, if colourings close to a split colouring are excluded.…

Complete multipartite graphs that are determined, up to switching, by their Seidel spectrum

It is known that complete multipartite graphs are determined by their distance spectrum but not by their adjacency spectrum. The Seidel spectrum of a graph $G$ on more than one vertex does not determine the graph, since any graph obtained from $G$ by Seidel switching has the same Seidel spectrum. We consider $G$ to be determined by its Seidel spectrum, up to switching, if any graph with the same spectrum is switching equivalent to a graph isomorphic to $G$. It is shown that any graph which has the same spectrum as a complete $k$-partite graph is switching equivalent to a complete $k$-partite g…

Reciprocal distance energy of complete multipartite graphs

In this paper, first we compute the energy of a special partitioned matrix under some cases. As a consequence, we obtain the reciprocal distance energy of the complete multipartite graph and also we give various other energies of complete multipartite graphs. Next, we show that among all complete k-partite graphs on n vertices, the complete split graph has minimum reciprocal distance energy and the reciprocal distance energy is maximum for the Turan graph. At last, it is shown that the reciprocal distance energy of the complete bipartite graph km,m decreases under deletion of an edge if 2\le m…

Some results on triangle partitions

We show that there exist efficient algorithms for the triangle packing problem in colored permutation graphs, complete multipartite graphs, distance-hereditary graphs, k-modular permutation graphs and complements of k-partite graphs (when k is fixed). We show that there is an efficient algorithm for C_4-packing on bipartite permutation graphs and we show that C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite graphs that have a triangle partition.

多重部超图的可选择性

Choosability of multipartite hypergraphs

本文研究了多重部超图的可选择性，证明了在特定条件下，此类超图可以被高效着色。

Cyclic hamiltonian cycle systems of the complete multipartite graph: even number of parts

A hamiltonian cycle system (HCS, for short) of a graph $Γ$ is a partition of the edges of $Γ$ into hamiltonian cycles. A HCS is cyclic when it is invariant under a cyclic permutation of all the vertices of $Γ$; the existence problem for a cyclic HCS has been completely solved by Buratti and Del Fra in 2004 when $Γ$ is the complete graph $K_v$, $v$ odd, and by Jordon and Morris in 2008 when $Γ$ is the complete graph minus a $1$-factor $K_v-I$, $v$ even. In this work we present a complete solution to the existence problem of a cyclic HCS for $Γ= K_{m\times n}$, the complete multipartite graph, w…

On the number of linear multipartite hypergraphs with given size

For any given integer $r\geqslant 3$, let $k=k(n)$ be an integer with $r\leqslant k\leqslant n$. A hypergraph is $r$-uniform if each edge is a set of $r$ vertices, and is said to be linear if two edges intersect in at most one vertex. Let $A_1,\ldots,A_k$ be a given $k$-partition of $[n]$ with $|A_i|=n_i\geqslant 1$. An $r$-uniform hypergraph $H$ is called {\it $k$-partite} if each edge $e$ satisfies $|e\cap A_i|\leqslant 1$ for $1\leqslant i\leqslant k$. In this paper, the number of linear $k$-partite $r$-uniform hypergraphs on $n\to\infty$ vertices is determined asymptotically when the numbe…

四边形的多色、多部拉姆齐数

Multicolor, multipartite Ramsey numbers for quadrilateral

本文研究四边形的多色多部拉姆齐数，给出一般上界及二色情形的全部精确值，并确定三部四色拉姆齐数为11。

关于涉及完全图的大小多部拉姆齐数

On the size multipartite Ramsey numbers involving complete graphs

该研究聚焦大小多部拉姆齐数m_j(H,G)，推导了基于H色数和G阶数的通用下界，给出大最大度图G对应的m_j(K_m,G)紧下界，还确定了m_j(K_m,K_{1,n})的数量级及多类特殊情形下的精确值。

Partitioning 3-edge-colored complete equi-bipartite graphs by monochromatic trees under a color degree condition

The monochromatic tree partition number of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum integer $k$ such that whenever the edges of $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $k$ vertex-disjoint monochromatic trees. In general, to determine this number is very difficult. For 2-edge-colored complete multipartite graph, Kaneko, Kano, and Suzuki gave the exact value of $t_2(K(n_1,n_2,...,n_k))$. In this paper, we prove that if $n\geq 3$, and K(n,n) is 3-edge-colored such that every vertex has color degree 3, then $t_3(K(n,n))=3.$

关于完全平衡多部图和分层图的直线交叉数

On the rectilinear crossing number of complete balanced multipartite graphs and layered graphs

本文研究完全平衡 r 部图 K_n^r 与相邻层全连接的分层图 L_n^r，通过构造直线画法给出二者直线交叉数的上界。

$(g,f)$-Chromatic spanning trees and forests

A heterochromatic (or rainbow) graph is an edge-colored graph whose edges have distinct colors, that is, where each color appears at most once. In this paper, I propose a $(g,f)$-chromatic graph as an edge-colored graph where each color $c$ appears at least $g(c)$ times and at most $f(c)$ times. I also present a necessary and sufficient condition for edge-colored graphs (not necessary to be proper) to have a $(g,f)$-chromatic spanning tree. Using this criterion, I show that an edge-colored complete graph $G$ has a spanning tree with a color probability distribution `similar' to that of $G$. Mo…

有向完全多部图中的英雄图

Heroes in oriented complete multipartite graphs

该研究针对有向图的英雄图刻画问题，证明拟传递有向图的英雄图与竞赛图一致，证否了有向多部图相关猜想，并给出有向完全多部图英雄图的近乎完整刻画。

关于约化词着色图的环面理想

On The Toric Ideals of the Coloured Graphs of Reduced Words

本文研究由阶梯分拆索引的伪多部图，给出其一般色多项式，并利用分拆恒等式构造与顶点正常着色相关的仿射环面理想，证明其射影版本可由卡通图实现。

Edge-colorings of graphs avoiding complete graphs with a prescribed coloring

Given a graph $F$ and an integer $r \ge 2$, a partition $\widehat{F}$ of the edge set of $F$ into at most $r$ classes, and a graph $G$, define $c_{r, \widehat{F}}(G)$ as the number of $r$-colorings of the edges of $G$ that do not contain a copy of $F$ such that the edge partition induced by the coloring is isomorphic to the one of $F$. We think of $\widehat{F}$ as the pattern of coloring that should be avoided. The main question is, for a large enough $n$, to find the (extremal) graph $G$ on $n$ vertices which maximizes $c_{r, \widehat{F}}(G)$. This problem generalizes a question of Erd{\H o}s…

$H$-分裂图上的极性

Polarity on $H$-split graphs

本文研究两类推广分裂图的图类上的 $(s,k)$-极性划分，给出判定算法及极小极性障碍阶数的上界。

完全多部图与Kneser型有向图的着色

Colouring Complete Multipartite and Kneser-type Digraphs

本文研究有向图双色数及其列表版本，推广了Lovász和Alon的结果，确定了Kneser图、Borsuk图及完全多部图的（列表）双色数渐近界，并给出Sabidussi定理的有向类比。

稠密多部图中偶环分解阈值

Even cycle decomposition thresholds for dense multipartite graphs

本文确定了稠密2-平衡多部图中偶环分解的阈值：对任意r≥2，C4分解阈值为2/3，C_{2k}（k≥3）分解阈值为1/2。

Laplacian $\{-1,0,1\}$- 和 $\{-1,1\}$-可对角化图

Laplacian $\{-1,0,1\}$- and $\{-1,1\}$-diagonalizable graphs

本文研究了能被 $\{-1,0,1\}$ 或 $\{-1,1\}$ 矩阵对角化的 Laplacian integral 图，引入 balanced 向量作为组合工具刻画其性质，并给出了 9 顶点以内可对角化连通图的完整列表。

关于独立横贯最大数的渐近紧界

An Asymptotically Sharp Bound on the Maximum Number of Independent Transversals

该研究针对给定边密度的多部图，证明了独立横贯最大数的渐近紧上界，推进了图论中独立横贯计数问题的研究。

Achievable multiplicity partitions in the inverse eigenvalue problem of a graph

Associated to a graph $G$ is a set $\mathcal{S}(G)$ of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be chosen. If $G$ has $n$ vertices, then the multiplicities of the eigenvalues of any matrix in $\mathcal{S}(G)$ partition $n$; this is called a multiplicity partition. We study graphs for which a multiplicity partition with only two integers is possible. The graphs $G$ for which there is a matrix in $\mathcal{S}(G)$ with partitions $[n-2,2]$ have been characteri…

Interval colorings of complete balanced multipartite graphs

A graph $G$ is called a complete $k$-partite ($k\geq 2$) graph if its vertices can be partitioned into $k$ independent sets $V_{1},...,V_{k}$ such that each vertex in $V_{i}$ is adjacent to all the other vertices in $V_{j}$ for $1\leq i<j\leq k$. A complete $k$-partite graph $G$ is a complete balanced $k$-partite graph if $|V_{1}| = |V_{2}| =... = |V_{k}|$. An edge-coloring of a graph $G$ with colors $1,...,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorabl…

Minimal obstructions to $2$-polar cographs

A graph is a cograph if it is $P_4$-free. A $k$-polar partition of a graph $G$ is a partition of the set of vertices of $G$ into parts $A$ and $B$ such that the subgraph induced by $A$ is a complete multipartite graph with at most $k$ parts, and the subgraph induced by $B$ is a disjoint union of at most $k$ cliques with no other edges. It is known that $k$-polar cographs can be characterized by a finite family of forbidden induced subgraphs, for any fixed $k$. A concrete family of such forbidden induced subgraphs is known for $k=1$, since $1$-polar graphs are precisely split graphs. For larger…

On Graph Isomorphism Problem

Let $G$ and $H$ be two simple graphs. A bijection $ϕ:V(G)\rightarrow V(H)$ is called an isomorphism between $G$ and $H$ if $(ϕv_i)(ϕv_j)\in E(H)$ $\Leftrightarrow$ $v_i v_j\in E(G)$, $\forall v_i,v_j \in V(G)$. In the case that $G = H$, we say $ϕ$ an automorphism of $G$ and denote the group consisting of all automorphisms of $G$ by $\mathrm{Aut}~G$. As well-known, the problem of determining whether or not two given graphs are isomorphic is called Graph Isomorphism Problem (GI). One of key steps in resolving GI is to work out the partition $Π^*_G$ of $V(G)$ composed of orbits of $\mathrm{Aut}~G…

Graph Varieties in High Dimension

We study the \emph{picture space} $X^d(G)$ of all embeddings of a finite graph $G$ as point-and-line arrangements in an arbitrary-dimensional projective space, continuing previous work on the planar case. The picture space admits a natural decomposition into smooth quasiprojective subvarieties called \emph{cellules}, indexed by partitions of the vertex set of $G$, and the irreducible components of $X^d(G)$ correspond to cellules that are maximal with respect to a partial order on partitions that is in general weaker than refinement. We study both general properties of this partial order and it…

Characterization of general position sets and its applications to cographs and bipartite graphs

A vertex subset $S$ of a graph $G$ is a general position set of $G$ if no vertex of $S$ lies on a geodesic between two other vertices of $S$. The cardinality of a largest general position set of $G$ is the general position number ${\rm gp}(G)$ of $G$. It is proved that $S\subseteq V(G)$ is in general position if and only if the components of $G[S]$ are complete subgraphs, the vertices of which form an in-transitive, distance-constant partition of $S$. If ${\rm diam}(G) = 2$, then ${\rm gp}(G)$ is the maximum of $ω(G)$ and the maximum order of an induced complete multipartite subgraph of the co…

The Locating Chromatic Number of the Join of Graphs

Let $f$ be a proper $k$-coloring of a connected graph $G$ and $Π=(V_1,V_2,...,V_k)$ be an ordered partition of $V(G)$ into the resulting color classes. For a vertex $v$ of $G$, the color code of $v$ with respect to $Π$ is defined to be the ordered $k$-tuple $c_{{}_Π}(v):=(d(v,V_1),d(v,V_2),...,d(v,V_k)),$ where $d(v,V_i)=\min\{d(v,x)|x\in V_i\}, 1\leq i\leq k$. If distinct vertices have distinct color codes, then $f$ is called a locating coloring. The minimum number of colors needed in a locating coloring of $G$ is the locating chromatic number of $G$, denoted by $\Cchi_{{}_L}(G)$. In this pap…

Erdős-Pyber theorem for hypergraphs and secret sharing

A new, constructive proof with a small explicit constant is given to the Erdős-Pyber theorem which says that the edges of a graph on $n$ vertices can be partitioned into complete bipartite subgraphs so that every vertex is covered at most $O(n/\log n)$ times. The theorem is generalized to uniform hypergraphs. Similar bounds with smaller constant value is provided for fractional partitioning both for graphs and for uniform hypergraphs. We show that these latter constants cannot be improved by more than a factor of 1.89 even for fractional covering by arbitrary complete multipartite subgraphs or…

Minimal obstructions to $(\infty, k)$-polarity in cographs

A graph is a cograph if it does not contain a 4-vertex path as an induced subgraph. An $(s, k)$-polar partition of a graph $G$ is a partition $(A, B)$ of its vertex set such that $A$ induces a complete multipartite graph with at most $s$ parts, and $B$ induces the disjoint union of at most $k$ cliques with no other edges. A graph $G$ is said to be $(s, k)$-polar if it admits an $(s, k)$-polar partition. The concepts of $(s, \infty)$-, $(\infty, k)$-, and $(\infty, \infty)$-polar graphs can be analogously defined. Ekim, Mahadev and de Werra pioneered in the research on polar cographs, obtaining…

Positive matching decompositions of graphs

A matching $M$ in a graph $Γ$ is positive if $Γ$ has a vertex-labeling such that $M$ coincides with the set of edges with positive weights. A positive matching decomposition (pmd) of $Γ$ is an edge-partition $M_1,\ldots,M_p$ of $Γ$ such that $M_i$ is a positive matching in $Γ-M_1\cup\cdots\cup M_{i-1}$, for $i=1,\ldots,p$. The pmds of graphs are used to study algebraic properties of the Lovász-Saks-Schrijver ideals arising from orthogonal representations of graphs. We give a characterization of pmds of graphs in terms of alternating closed walks and apply it to study pmds of various classes of…

Minimal obstructions to $(s,1)$-polarity in cographs

Let $k,l$ be nonnegative integers. A graph $G$ is $(k,l)$-polar if its vertex set admits a partition $(A,B)$ such that $A$ induces a complete multipartite graph with at most $k$ parts, and $B$ induces a disjoint union of at most $l$ cliques with no other edges. A graph is a cograph if it does not contain $P_4$ as an induced subgraph. It is known that $(k,l)$-polar cographs can be characterized through a finite family of forbidden induced subgraphs, for any fixed choice of $k$ and $l$. The problem of determining the exact members of such family for $k = 2 = l$ was posted by Ekim, Mahadev and de…

k-tuple total restrained domination and k-tuple total restrained domatic in graphs

Let $G$ be a graph of order $n$ and size $m$ and let $k\geq 1$ be an integer. A $k$-tuple total dominating set in $G$ is called a $k$-tuple total restrained dominating set of $G$ if each vertex $x\in V(G)-S$ is adjacent to at least $k$ vertices of $V(G)-S$. The minimum number of vertices of a such sets in $G$ are the $k$-tuple total restrained domination number $γ_{\times k,t}^{r}(G)$ of $G$. The maximum number of classes of a partition of $V(G)$ such that its all classes are $k$-tuple total restrained dominating sets in $G$, is called the $k$-tuple total restrained domatic number of $G$. In t…