Andean E. Medjedovic
We solve a problem conjectured by Cheriyan, giving sharp bounds for incidence of certain edge partitions of the connected graph on $n$-vertices. We briefly discuss the history of the problem and relation to node connectivity of strongly regular graphs. We show that the bound cannot be made sharper.
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Partitioning the projective plane to two incidence-rich parts
受高围长图内部划分存在性证明启发，本文为射影平面关联图的内部划分提供了构造性证明，探讨了其几何性质，并精确计算了平方阶Desarguesian平面关联图中各类邻居集大小的最大差异。
Incidence-free sets and edge domination in incidence graphs
本文研究了 incidence 图的边支配数，证明在对称设计情况下，该问题等价于确定最大无 incidence 点集的大小。
An oriented hypergraph is a hypergraph together with an incidence orientation such that each edge-vertex incidence is given a label of $+1$ or $-1$. An oriented hypergraph is called incidence balanced if there exists a bipartition of the vertex set such that every edge intersects one part of the bipartition in positively incident vertices with the edge and other part in negatively incident vertices with the edge. In this paper, we investigate the incidence balance of oriented hypergraphs and induced signed hypergraphs by means of the eigenvalues of associated matrices and tensors, and provide …
On the Incidence matrices of hypergraphs
本研究分析超图的边-顶点及顶点-边关联矩阵的秩和零空间，发现k-均匀圈、单位等结构影响零空间，单位收缩保持秩不变，并建立与邻接矩阵特征值的联系。
Certified Countermodels in Profile and Incidence Fibres of Diamond-Induced Edge Partitions
该研究针对有向图的钻石诱导典范边划分，探究关系凭证的标识能力，通过证明划分性质、计算纤维与邻域、构造反例，得出凭证具有中等选择性的结论，结果可独立验证。
We consider a space of sparse Boolean matrices of size $n \times n$, which have finite co-rank over $GF(2)$ with high probability. In particular, the probability such a matrix has full rank, and is thus invertible, is a positive constant with value about $0.2574$ for large $n$. The matrices arise as the vertex-edge incidence matrix of 1-out 3-uniform hypergraphs The result that the null space is finite, can be contrasted with results for the usual models of sparse Boolean matrices, based on the vertex-edge incidence matrix of random $k$-uniform hypergraphs. For this latter model, the expected …
We determine the partition dimension of the incidence graph $G(Π_q)$ of the projective plane $Π_q$ up to a constant factor $2$ as $(2+o(1))\log_2{q}\leq \mathrm{pd}(G(Π_q))\leq (4+o(1))\log_2{q}.$
Kang and Park recently showed that every cubic (loopless) multigraph is incidence 6-choosable [On incidence choosability of cubic graphs. \emph{arXiv}, April 2018]. Equivalently, every bipartite graph obtained by subdividing once every edge of a cubic graph, is strongly 6-edge-choosable. The aim of this note is to give a shorter proof of their result by looking at the strong edge-coloring formulation of the problem.
We prove bounds on the number of incidences between a set of algebraic curves in $\mathbb{C}^2$ and a Cartesian product $A\times B$ with finite sets $A,B\subset \mathbb{C}$. Similar bounds are known under various conditions, but we show that the Cartesian product assumption leads to a simpler proof. This assumption holds in a number of interesting applications, and with our bound these applications can be extended from $\mathbb{R}$ to $\mathbb{C}$. The proof is a new application of the polynomial partitioning technique introduced by Guth and Katz.
An incidence of a graph $G$ is a pair $(u,e)$ where $u$ is a vertex of $G$ and $e$ is an edge of $G$ incident with $u$. Two incidences $(u,e)$ and $(v,f)$ of $G$ are adjacent whenever (i) $u=v$, or (ii) $e=f$, or (iii) $uv=e$ or $uv=f$. An incidence $k$-coloring of $G$ is a mapping from the set of incidences of $G$ to a set of $k$ colors such that every two adjacent incidences receive distinct colors. The notion of incidence coloring has been introduced by Brualdi and Quinn Massey (1993) from a relation to strong edge coloring, and since then, attracted by many authors. On a list version of in…
Spectral Applications of Vertex-Clique Incidence Matrices Associated with a Graph
该研究引入图的顶点-团关联矩阵，推导图负特征值、负惯性指数及相关能量的界，推广无符号拉普拉斯矩阵等概念，并借此证明多类满足q(G)=2的图，建立了团划分、边团覆盖与图谱的关联。
The vertex-edge incidence matrix of a (connected) unicyclic graph G is a square matrix which is invertible if and only if the cycle of G is an odd cycle. A combinatorial formula of the inverse of the incidence matrix of an odd unicyclic graph was known. A combinatorial formula of the Moore-Penrose inverse of the incidence matrix of an even unicyclic graph is presented solving an open problem.
Asymptotic number of edge-colored regular graphs
本文推导了边着色正则图的渐近数量公式，并应用该结果计算了大随机k-正则图的c-边着色期望数量。
We adapt here the computation of characters on incidence Hopf algebras introduced by W. Schmitt in the 1990s to a family mixing bounded and unbounded posets. We then apply our results to the family of hypertree posets and partition posets. As a consequence, we obtain some enumerative formulas and a new proof for the computation of the Moebius numbers of the hypertree posets. Moreover, we compute the coproduct of the incidence Hopf algebra and recover a known formula for the number of hypertrees with fixed valency set and edge sizes set.
Linear Tur'an Numbers of Four-Edge Uniform Paths via Incidence Rank
本文通过关联秩不等式证明了四边均匀路径的线性Turán数猜想，给出边数上界$(r+1)n/r$，并刻画了等号条件为Steiner系统$S(2,r,r^2)$的并集。
Fuzzy incidence graphs (FIG) model real world problems efficiently when there is an extra attribute of vertex-edge relationship. The article discusses the operations on Fuzzy incidence graphs. The join, Cartesian product, tensor product, and composition of FIGs are explored. The study is concentrated mainly on strong fuzzy incidence graphs (SFIG). The idea of strong incidence domination (SID) is used, and strong incidence domination number (SIDN) in operations is examined. Basic properties of FIGs obtained from the operations are studied. Bounds for the domination number of product of two SFIG…
We investigate extremal functions ex_e(F,n) and ex_i(F,n) counting maximum numbers of edges and maximum numbers of vertex-edge incidences in simple hypergraphs H which have n vertices and do not contain a fixed hypergraph F; the containment respects linear orderings of vertices. We determine both functions exactly if F has only distinct singleton edges or if F is one of the 55 hypergraphs with at most four incidences (we give proofs only for six cases). We prove some exact formulae and recurrences for the numbers of hypergraphs, simple and all, with n incidences and derive rough logarithmic as…
An incidence in a graph $G$ is a pair $(v,e)$ where $v$ is a vertex of $G$ and $e$ is an edge of $G$ incident to $v$. Two incidences $(v,e)$ and $(u,f)$ are adjacent if at least one of the following holds: $(a)$ $v = u$, $(b)$ $e = f$, or $(c)$ $vu \in \{e,f\}$. An incidence coloring of $G$ is a coloring of its incidences assigning distinct colors to adjacent incidences. It was conjectured that at most $Δ(G) + 2$ colors are needed for an incidence coloring of any graph $G$. The conjecture is false in general, but the bound holds for many classes of graphs. We introduce some sufficient properti…
The ordinary generating function of the number of complete subgraphs (cliques) of $G$, denoted by $C(G,x)$, is called the The clique polynomial of the graph $G$. In this paper, we first introduce some \emph{clique} incidence matrices associated by a simple graph $G$ as a generalization of the classical vertex-edge incidence matrix of $G$. Then, using these clique incidence matrices, we obtain two clique-counting identities that can be used for deriving two combinatorial formulas for the first and the second derivatives of clique polynomials. Finally, we conclude the paper with several open que…
The determinants of $\{\pm 1\}$-matrices are calculated by via the oriented hypergraphic Laplacian and summing over an incidence generalization of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on their hyperedge containment. Every non-edge-monic family is shown to contribute a net value of $0$ to the Laplacian, while each edge-monic family is shown to sum to the absolute value of the determinant of the original incidence matrix. Simple symmetries are identified as well as their relationship to Hadamard's maximum determinant problem. Finally, the entries…
The Heawood graph is the point-block incidence graph of the Fano plane (the unique Steiner triple system of order 7). We show that the minimum semidefinite rank of this graph is 10. That is, 10 is the smallest number of complex dimensions in which this graph has a faithful orthogonal representation, i.e., an assignment of a vector to each vertex such that the edges occur between precisely those vertices given non-orthogonal pairs. Some of our techniques extend to the incidence graphs of Steiner triple systems of larger order, and we include some observations and questions about the more genera…
We show that for every l, there exists d_l such that every 3-edge-connected graph with minimum degree d_l can be edge-partitioned into paths of length l (provided that its number of edges is divisible by l). This improves a result asserting that 24-edge-connectivity and high minimum degree provides such a partition. This is best possible as 3-edge-connectivity cannot be replaced by 2-edge connectivity.
Cyclic Incidence Orderings of Complete Graphs and 3-Uniform Hypergraphs
本文研究完全k-一致超图边的循环排序，其中顶点关联序列为公共词的循环移位，证明了k=2和k=3时排序存在的充要条件，并利用反射卷积和配对相交计数完成必要性证明。
Let $G$ be a simple graph with vertex set $V(G) = \{v_1, v_2,\ldots, v_n\}$ and edge set $E(G) = \{e_1, e_2,\ldots, e_m\}$. Similar to the Randić matrix, here we introduce the Randić incidence matrix of a graph $G$, denoted by $I_R(G)$, which is defined as the $n\times m$ matrix whose $(i, j)$-entry is $(d_i)^{-\frac{1}{2}}$ if $v_i$ is incident to $e_j$ and $0$ otherwise. Naturally, the Randić incidence energy $I_RE$ of $G$ is the sum of the singular values of $I_R(G)$. We establish lower and upper bounds for the Randić incidence energy. Graphs for which these bounds are best possible are cha…
Abstract Orientable Incidence Structure and Algorithms for Finite Bounded Acyclic Categories. II. Data Structure and Fundamental Operations
本文构建了有限有界无环范畴的数据结构以编码抽象可定向关联结构，将其表示为带权有向无环多重图，并从几何、范畴和编程角度研究了其基本操作。
This paper considers the difficulty in the set-system approach to generalizing graph theory. These difficulties arise categorically as the category of set-system hypergraphs is shown not to be cartesian closed and lacks enough projective objects, unlike the category of directed multigraphs (i.e. quivers). The category of incidence hypergraphs is introduced as a "graph-like" remedy for the set-system issues so that hypergraphs may be studied by their locally graphic behavior via homomorphisms that allow an edge of the domain to be mapped into a subset of an edge in the codomain. Moreover, it is…
An incidence of a graph $G$ is a vertex-edge pair $(v,e)$ such that $v$ is incidence with $e$. A conflict-free incidence coloring of a graph is a coloring of the incidences in such a way that two incidences $(u,e)$ and $(v,f)$ get distinct colors if and only if they conflict each other, i.e.,(i) $u=v$, (ii) $uv$ is $e$ or $f$, or (iii) there is a vertex $w$ such that $uw=e$ and $vw=f$. The minimum number of colors used among all conflict-free incidence colorings of a graph is the conflict-free incidence chromatic number. A graph is outer-1-planar if it can be drawn in the plane so that vertice…
We construct an involution on set partitions which keeps track of the numbers of crossings, nestings and alignments of two edges. We derive then the symmetric distribution of the numbers of crossings and nestings in partitions, which generalizes Klazar's recent result in perfect matchings. By factorizing our involution through bijections between set partitions and some path diagrams we obtain the continued fraction expansions of the corresponding ordinary generating functions.
In response to a well-known open question ``Does every complete geometric graph on $2n\/$ vertices have a partition of its edge set into $n\/$ plane spanning trees?" we provide an affirmative answer when the complete geometry graph is in the regular wheel configuration. Also we present sufficient conditions for the complete geometric graph on $2n\/$ vertices to have a partition of its edge set into $n\/$ plane spanning trees (which are double stars, caterpillars or $ w\/$-caterpillars).
Interplay of Cubic Graphs and Simplicial Surfaces
本文研究单纯曲面与三次图性质的相互转化，探讨三次图在单纯曲面上的嵌入及其与强图嵌入的联系，并刻画了单纯球面使三次图能嵌入非负Euler示性数曲面的性质。
Let $G=(V,E)$ be a graph. A set of vertices $A$ is an incidence generator for $G$ if for any two distinct edges $e,f\in E(G)$ there exists a vertex from $A$ which is an endpoint of either $e$ or $f$. The smallest cardinality of an incidence generator for $G$ is called the incidence dimension and is denoted by $dim_I(G)$. A set of vertices $P$ is a 2-packing if the distance between any pair of distinct vertices from $P$ is greater than two. The largest cardinality of a 2-packing of $G$ is the packing number of $G$ and is denoted by $ρ(G)$. The incidence dimension of graphs is introduced and stu…
An edge colouring of a graph is said to be an $r$-local colouring if the edges incident to any vertex are coloured with at most $r$ colours. Generalising a result of Bessy and Thomassé, we prove that the vertex set of any $2$-locally coloured complete graph may be partitioned into two disjoint monochromatic cycles of different colours. Moreover, for any natural number $r$, we show that the vertex set of any $r$-locally coloured complete graph may be partitioned into $O(r^2 \log r)$ disjoint monochromatic cycles. This generalises a result of Erdős, Gyárfás and Pyber.
Chromatic Index of Signed Generalized Book Graphs and Signed Complete Graphs
本文研究了符号广义书图和阶数不超过六的符号完全图的色数问题，通过分析符号图的边着色规则，确定了这两种图类的色数。
The vertices of any graph with $m$ edges may be partitioned into two parts so that each part meets at least $\frac{2m}{3}$ edges. Bollobás and Thomason conjectured that the vertices of any $r$-uniform hypergraph with $m$ edges may likewise be partitioned into $r$ classes such that each part meets at least $\frac{r}{2r-1}m$ edges. In this paper we prove the weaker statement that, for each $r\ge 4$, a partition into $r$ classes may be found in which each class meets at least $\frac{r}{3r-4}m$ edges, a substantial improvement on previous bounds.
A {\it heterochromatic tree} is an edge-colored tree in which any two edges have different colors. The {\it heterochromatic tree partition number} of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum positive integer $p$ such that whenever the edges of the graph $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $p$ vertex-disjoint heterochromatic trees. In this paper we determine the heterochromatic tree partition number of an $r$-edge-colored complete graph.
Sharp bounds on the least eigenvalue of an arbitrary graph are presented. Necessary and sufficient (just sufficient) conditions for the lower (upper) bound to be attained are deduced using edge clique partitions. As an application, we prove that the least eigenvalue of the $n$-Queens' graph $\mathcal{Q}(n)$ is equal to $-4$ for every $n \ge 4$ and it is also proven that the multiplicity of this eigenvalue is $(n-3)^2$. Additionally, some results on the edge clique partition graph parameters are obtained.
The Barát-Thomassen conjecture asserts that there is a function $f$ such that for every fixed tree $T$ with $t$ edges, every graph which is $f(t)$-edge-connected with its number of edges divisible by $t$ has a partition of its edges into copies of $T$. This has been proved in the case of paths of length $2^k$ by Thomassen, and recently shown to be true for all paths by Botler, Mota, Oshiro and Wakabayashi. Our goal in this paper is to propose an alternative proof of the path case with a weaker hypothesis: Namely, we prove that there is a function $f$ such that every $24$-edge-connected graph w…
Positional games have been introduced by Hales and Jewett in 1963 and have been extensively investigated in the literature since then. These games are played on a hypergraph where two players alternately select an unclaimed vertex of it. In the Maker-Breaker convention, if Maker manages to fully take a hyperedge, she wins, otherwise, Breaker is the winner. In the Maker-Maker convention, the first player to take a hyperedge wins. In both cases, the game stops as soon as Maker has taken a hyperedge. By definition, this family of games does not handle scores and cannot represent games in which pl…
In 1978, Richard Rado showed that every edge-coloured complete graph of countably infinite order can be partitioned into monochromatic paths of different colours. He asked whether this remains true for uncountable complete graphs and a notion of \emph{generalised paths}. In 2016, Daniel Soukup answered this in the affirmative and conjectured that a similar result should hold for complete bipartite graphs with bipartition classes of the same infinite cardinality, namely that every such graph edge-coloured with $r$ colours can be partitioned into $2r-1$ monochromatic generalised paths with each …
For a given hypergraph, an orientation can be assigned to the vertex-edge incidences. This orientation is used to define the adjacency and Laplacian matrices. In addition to studying these matrices, several related structures are investigated including the incidence dual, the intersection graph (line graph), and the 2-section. A connection is then made between oriented hypergraphs and balanced incomplete block designs.
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 $t_3(K(n,n))=3.$
Simplification & Incidence: How an Incidence-focused Perspective Patches Category-theoretic Problems in Graph Theory
本文从关联视角出发，通过弱集合系统同态修复了图操作在传统同态下丧失函子性的三个范畴论缺陷，并证明其范畴与关联结构范畴同构。
A topological graph is a graph drawn in the plane. A topological graph is $k$-plane, $k>0$, if each edge is crossed at most $k$ times. We study the problem of partitioning the edges of a $k$-plane graph such that each partite set forms a graph with a simpler structure. While this problem has been studied for $k=1$, we focus on optimal $2$-plane and $3$-plane graphs, which are $2$-plane and $3$-plane graphs with maximum density. We prove the following results. (i) It is not possible to partition the edges of a simple optimal $2$-plane graph into a $1$-plane graph and a forest, while (ii) an edg…
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。
The signless Laplacian Q and signless edge-Laplacian S of a given graph may or may not be invertible. The Moore-Penrose inverses of Q and S are studied. In particular, using the incidence matrix, we find combinatorial formulas of the Moore- Penrose inverses of Q and S for trees. Also we present combinatorial formulas of the inverses of Q and S for odd unicyclic graphs.
An incidence of a graph G is a pair (v, e) where v is a vertex of G and e is an edge of G incident with v. Two incidences (v, e) and (w, f) of G are adjacent whenever (i) v = w, or (ii) e = f , or (iii) vw = e or f. An incidence p-colouring of G is a mapping from the set of incidences of G to the set of colours {1,. .. , p} such that every two adjacent incidences receive distinct colours. Incidence colouring has been introduced by Brualdi and Quinn Massey in 1993 and, since then, studied by several authors. In this paper, we introduce and study the list version of incidence colouring. We deter…
In this paper, we begin by partitioning the edges (or arcs) of a circulant (di)graph according to which generator in the connection set leads to each edge. We then further refine the partition by subdividing any part that corresponds to an element of order less than $n$, according to which of the cycles generated by that element the edge is in. It is known that if the (di)graph is connected and has no multiple edges, then any automorphism that respects the first partition and fixes the vertex corresponding to the group identity must be an automorphism of the group (this is in fact true in the …
Let $π$ and $λ$ be two set partitions with the same number of blocks. Assume $π$ is a partition of $[n]$. For any integer $l, m \geq 0$, let $\mathcal{T}(π, l)$ be the set of partitions of $[n+l]$ whose restrictions to the last $n$ elements are isomorphic to $π$, and $\mathcal{T}(π, l, m)$ the subset of $\mathcal{T}(π,l)$ consisting of those partitions with exactly $m$ blocks. Similarly define $\mathcal{T}(λ, l)$ and $\mathcal{T}(λ, l,m)$. We prove that if the statistic $cr$ ($ne$), the number of crossings (nestings) of two edges, coincides on the sets $\mathcal{T}(π, l)$ and $\mathcal{T}(λ, l…
Combinatorial formula for the Moore-Penrose inverse of the complex signless Laplacian of an oriented graph
研究具有非零复边权重的弱连通有向图，通过找到无符号关联矩阵秩的充要条件，得出复无符号关联矩阵和复无符号拉普拉斯矩阵的摩尔 - 彭罗斯逆的组合公式，解决了相关开放问题。
Edge partitions into induced-$2K_2$-free bipartite graphs
本文研究了将双图边划分到诱导-2K₂-自由双图的问题，确定了最小部分数并探讨了相关界限和复杂性。
Component order edge connectivity, vertex degrees, and integer partitions
针对NP难的图的k-分量阶边连通度求解问题，研究推导了顶点度条件以给出该连通度的下界，分析了不同下界值的条件生成复杂度，并证明了相关整数分拆结果。
Monochromatic cycle partitions in 3-mean edge-colourings
针对Conlon与Stein提出的边着色完全图划分为有限个单色圈的问题，证明顶点关联颜色平均数至多为3时该命题成立，解决了r=3的首个公开情况。
On several judicious partitioning problems of graphs and hypergraphs
本文针对图与超图的明智划分问题，证明了Bollobás与Scott、Bollobás与Thomason的多个猜想，得到了最优的划分边数下界。
The vertices of any graph with $m$ edges can be partitioned into two parts so that each part meets at least $\frac{2m}{3}$ edges. Bollobás and Thomason conjectured that the vertices of any $r$-uniform graph may be likewise partitioned into $r$ classes such that each part meets at least $cm$ edges, with $c=\frac{r}{2r-1}$. In this paper, we prove this conjecture for the case $r=3$. In the course of the proof we shall also prove an extension of the graph case which was conjectured by Bollobás and Scott.
We answer a question of Gyárfás and Sárközy from 2013 by showing that every 2-edge-coloured complete 3-uniform hypergraph can be partitioned into two monochromatic tight paths of different colours. We also give a lower bound for the number of tight paths needed to partition any 2-edge-coloured complete r-partite r-uniform hypergraph. Finally, we show that any 2-edge-coloured complete bipartite graph has a partition into a monochromatic cycle and a monochromatic path, of different colours, unless the colouring is a split colouring.
An incidence of a graph $G$ is a pair $(v,e)$ where $v$ is a vertex of $G$ and $e$ an edge incident to $v$. Two incidences $(v,e)$ and $(w,f)$ are adjacent whenever $v = w$, or $e = f$, or $vw = e$ or $f$. The incidence coloring game [S.D. Andres, The incidence game chromatic number, Discrete Appl. Math. 157 (2009), 1980-1987] is a variation of the ordinary coloring game where the two players, Alice and Bob, alternately color the incidences of a graph, using a given number of colors, in such a way that adjacent incidences get distinct colors. If the whole graph is colored then Alice wins the g…
Almost partitioning every $2$-edge-coloured complete $k$-graph into $k$ monochromatic tight cycles
研究红蓝着色完全$k$-图的单色紧圈划分问题，证明其中存在$k$个顶点互不相交的单色紧圈，可覆盖除$o(n)$外的全部顶点。
We present results on partitioning the vertices of $2$-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of Sárközy: the vertex set of every $2$-edge-colored graph can be partitioned into at most $2α(G)$ monochromatic cycles, where $α(G)$ denotes the independence number of $G$. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from $o(|V(G)|)$ vertices, the vertex set of any $2$-edge-colored graph $G$ with minimum degree at least $(1…
An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. We define the adjacency, incidence and Laplacian matrices of an oriented hypergraph and study each of them. We extend several matrix results known for graphs and signed graphs to oriented hypergraphs. New matrix results that are not direct generalizations are also presented. Finally, we study a new family of matrices that contains walk information.
This article provides structural characterization of simple graphs whose edge-set can be partitioned into maximum matchings. We use Vizing's classification of simple graphs based on edge chromatic index.
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