The Inductive Graph Dimension from The Minimum Edge Clique Cover
Kassahun Betre, Evatt Salinger
备注：
arXiv admin note: substantial text overlap with arXiv:1903.02523 author note: this version was meant to be a replacement of arXiv:1903.02523, but due to submission mistake it appeared as a new submission
In this paper we prove that the inductively defined graph dimension has a simple additive property under the join operation. The dimension of the join of two simple graphs is one plus the sum of the dimensions of the component graphs: $\mathrm{dim}\, (G_1+ G_2) = 1 +\mathrm{dim}\, G_1+ \mathrm{dim}\, G_2$. We use this formula to derive an expression for the inductive dimension of an arbitrary finite simple graph from its minimum edge clique cover. A corollary of the formula is that any arbitrary finite simple graph whose maximal cliques are all of order $N$ has dimension $N-1$. We finish by finding lower and upper bounds on the inductive dimension of a simple graph in terms of its clique number.
In this paper we prove that the recursive (Knill) dimension of the join of two graphs has a simple formula in terms of the dimensions of the component graphs: $\mathrm{dim\,} (G_1+G_2) = 1 +\mathrm{dim\,} G_1+ \mathrm{dim\,} G_2$. We use this formula to derive an expression for the Knill dimension of a graph from its minimum clique cover. A corollary of the formula is that a graph made of the arbitrary union of complete graphs $K_N$ of the same order $N$ will have dimension $N-1$. We finish by finding lower and upper bounds on the Knill dimension of a graph in terms of its clique number.
An edge clique cover of a graph is a set of cliques that covers all edges of the graph. We generalize this concept to "$K_t$ clique cover", i.e. a set of cliques that covers all complete subgraphs on $t$ vertices of the graph, for every $t \geq 1$. In particular, we extend a classical result of Erdös, Goodman, and Pósa (1966) on the edge clique cover number ($t = 2$), also known as the intersection number, to the case $t = 3$. The upper bound is tight, with equality holding only for the Turán graph $T(n,3)$. We also extend an algorithm of Scheinerman and Trenk (1999) to solve a weighted versio…
This work examines the problem of clique enumeration on a graph by exploiting its clique covers. The principle of inclusion/exclusion is applied to determine the number of cliques of size $r$ in the graph union of a set $\mathcal{C} = \{c_1, \ldots, c_m\}$ of $m$ cliques. This leads to a deeper examination of the sets involved and to an orbit partition, $Γ$, of the power set $\mathcal{P}(\mathcal{N}_{m})$ of $\mathcal{N}_{m} = \{1, \ldots, m\}$. Applied to the cliques, this partition gives insight into clique enumeration and yields new results on cliques within a clique cover, including expres…
It takes $n^2/4$ cliques to cover all the edges of a complete bipartite graph $K_{n/2,n/2}$, but how many cliques does it take to cover all the edges of a graph $G$ if $G$ has no $K_{t,t}$ induced subgraph? We prove that $O(|G|^{2-1/(2t)})$ cliques suffice; and also prove that, even for graphs with no stable set of size four, we may need more than linearly many cliques. This settles two questions discussed at a recent conference in Lyon.
For a $k$-uniform hypergraph $F$ we consider the parameter $Θ(F)$, the minimum size of a clique cover of the of $F$. We derive bounds on $Θ(F)$ for $F$ belonging to various classes of hypergraphs.
The edge clique cover number $ecc(G)$ of a graph $G$ is the size of the smallest set of complete subgraphs whose union covers all edges of $G$. It has been conjectured that all the simple graphs with independence number two satisfy $ecc(G)\leq n$. First, we present a class of graphs containing edges difficult to cover but that satisfy the conjecture. Second, we describe a large class of graphs $G$ such that $ecc(G)\leq \frac{3}{2}n$. This class is easy to characterize.
Let $G$ be a 2-connected $n$-vertex graph and $N_s(G)$ be the total number of $s$-cliques in $G$. Let $k\ge 4$ and $s\ge 2$ be integers. In this paper, we show that if $G$ has an edge $e$ which is not on any cycle of length at least $k$, then $N_s(G)\le r{k-1\choose s}+{t+2\choose s}$, where $n-2=r(k-3)+t$ and $0\le t\le k-4$. This result settles a conjecture of Ma and Yuan and provides a clique version of a theorem of Fan, Wang and Lv. As a direct corollary, if $N_s(G)> r{k-1\choose s}+{t+2\choose s}$, every edge of $G$ is covered by a cycle of length at least $k$.
We show that a graph with $n$ vertices and vertex cover of size $k$ has at most $4^k + n$ potential maximal cliques. We also show that for each positive integer $k$, there exists a graph with vertex cover of size $k$, $O(k^2)$ vertices, and $Ω(4^k)$ potential maximal cliques. Our results extend the results of Fomin, Liedloff, Montealegre, and Todinca [Algorithmica, 80(4):1146--1169, 2018], who proved an upper bound of $poly(n) 4^k$, but left the lower bound as an open problem.
On Intersection Representations and Clique Partitions of Graphs
有限简单图$G$的多族集合表示是集合的多族$\mathcal{F}$（不一定是不同的），其中每个集合表示$G$中的一个顶点，并且$\mathcal{F}$中的两个集合相交当且仅当两个对应的顶点相邻。对于图$G$，\textit{edge clique covering}（分别为\textit{edge clique partition}）$\mathcal{Q}$是一组团，其中每条边都包含在$\mathcal{Q}$的一个成员\textit{at least}（分别为\textit{exactly}）中。1966年，P. Erdös，A. Goodman和L. Pósa（The representation of a graph by set intercussions，\textit{Canadian J. Math.}，\textbf{18}，pp.106-112）指出，对于一个图，多族集合表示$\mathcal{F}$和边集的团覆盖$\mathcal{Q}$之间存在一一对应关系。此外，对于一个图，一个可以类似地具有一对一的对应关系之间的特定的多族集表示与相交大小最多为一和团分区的边缘集。1990年，S. McGuinness和R. Rees（On the number of distinct minimal club partitions and club covers of…
We study the relationship between the eternal domination number of a graph and its clique covering number using both large-scale computation and analytic methods. In doing so, we answer two open questions of Klostermeyer and Mynhardt. We show that the smallest graph having its eternal domination number less than its clique covering number has $10$ vertices. We determine the complete set of $10$-vertex and $11$-vertex graphs having eternal domination numbers less than their clique covering numbers. We show that the smallest triangle-free graph with this property has order $13$, as does the smal…
An exact upper bound on the size of minimal clique covers
不定字符串在最近的过去受到了相当大的关注;例如，参见Christodoulakis et al 2015和Helling et al 2017。这种关注是由于它们在生物信息学中的适用性，以及与无向图的自然对应。这种对应的一个方面是这样一个事实，即表示任何给定的无向图的不定数的最小字母表大小对应于该图的最小团覆盖的大小。本文解决了Helling等人在2017年提出的一个相关问题：计算$Θ_n(m)$，它是最大可能的最小团覆盖的大小（即，精确的上限），以及相应的不确定性的字母表大小，任何图的$n$顶点和$m$边。
Extremal problems on disjoint path covers of graphs
School of Mathematics and Statistics, Key Laboratory of Analytical Mathematics and Applications (Ministry of Education), Fujian Key Laboratory of Analytical Mathematics and Applications (FJKLAMA), Center for Applied Mathematics of Fujian Province (FJNU), Fujian Normal University（福建师范大学数学与统计学院，分析数学及应用教育部重点实验室，福建省分析数学及应用重点实验室（FJKLAMA），福建省应用数学中心（FJNU））
For a graph $G$, let $cp(G)$ denote the minimum number of cliques of $G$ needed to cover the edges of $G$ exactly once. Similarly, let $bp_k(G)$ denote the minimum number of bicliques (i.e. complete bipartite subgraphs of $G$) needed to cover each edge of $G$ exactly $k$ times. We consider two conjectures -- one regarding the maximum possible value of $cp(G) + cp(\overline{G})$ (due to de Caen, Erdős, Pullman and Wormald) and the other regarding $bp_k(K_n)$ (due to de Caen, Gregory and Pritikin). We disprove the first, obtaining improved lower and upper bounds on $\max_G cp(G) + cp(\overline{G…
The Lovász Theta Function for Recovering Planted Clique Covers and Graph Colorings
University of Southern California（南加州大学）；National University of Singapore（新加坡国立大学）；Singapore University of Technology and Design（新加坡科技设计大学）；Archimedes/Athena RC, Greece（希腊阿基米德/雅典研究中心）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）
The biclique cover number $(\text{bc})$ of a graph $G$ denotes the minimum number of complete bipartite (biclique) subgraphs to cover all the edges of the graph. In this paper, we show that $\text{bc}(G) \geq \lceil \log_2(\text{mc}(G^c)) \rceil \geq \lceil \log_2(χ(G)) \rceil$ for an arbitrary graph $G$, where $χ(G)$ is the chromatic number of $G$ and $\text{mc}(G^c)$ is the number of maximal cliques of the complementary graph $G^c$, i.e., the number of maximal independent sets of $G$. We also show that $\lceil \log_2(\text{mc}(G^c)) \rceil$ could be a strictly tighter lower bound of the bicl…
