For each m>=1 and k>=2, we construct a graph G=(V,E) with ω(G)=m such that max_{1\leq i\leq k} ω(G[V_i])=m for arbitrary partition V=V_1\cup...\cup V_k, where ω(G) is the clique number of G and G[V_i] is the induced subgraph of G with the vertex set V_i. Using this result, we show that for each m>=2 there exists an exact m-cover of Z which is not the union of two 1-covers.
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…
All the work made so far on edge-covering a graph by cliques focus on finding the minimum number of cliques that cover the graph. On this paper, we fix the number of cliques that cover a graph by the same number of vertices that the graph has, and give an upper bound for the sum of the number of vertices of these cliques in the cases where this covering is possible.
A clique covering of a graph $G$ is a set of cliques of $G$ such that any edge of $G$ is contained in one of these cliques, and the weight of a clique covering is the sum of the sizes of the cliques in it. The sigma clique cover number $scc(G)$ of a graph $G$, is defined as the smallest possible weight of a clique covering of $G$. Let $ K_t(d) $ denote the complete $ t $-partite graph with each part of size $d$. We prove that for any fixed $d \ge 2$, we have $$\lim_{t \rightarrow \infty} scc(K_t(d))= \frac{d}{2} t\log t.$$ This disproves a conjecture of Davoodi, Javadi and Omoomi.
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.
Let $\cal C$ be a clique covering for $E(G)$ and let $v$ be a vertex of $G$. The valency of vertex $v$ (with respect to $\cal C$), denoted by $val_{\cal C}(v)$, is the number of cliques in $\cal C$ containing $v$. The local clique cover number of $G$, denoted by $lcc(G)$, is defined as the smallest integer $k$, for which there exists a clique covering for $E(G)$ such that $val_{\cal C}(v)$ is at most $k$, for every vertex $v\in V(G)$. In this paper, among other results, we prove that if $G$ is a claw-free graph, then $lcc(G)+χ(G)\leq n+1$.
Given a weighted graph $G(V,E)$ with weight $\mathbf w: E\rightarrow Z^{|E|}_{+}$. A $k-$cycle covering is an edge subset $A$ of $E$ such that $G-A$ has no $k-$cycle. The minimum weight of $k-$cycle covering is the weighted covering number on $k-$cycle, denoted by $τ_{k}(G_{w})$. In this paper, we design a $k-1/2$ approximation algorithm for the weighted covering number on $k-$cycle when $k$ is odd. Given a weighted graph $G(V,E)$ with weight $\mathbf w: E\rightarrow Z^{|E|}_{+}$. A $k-$clique covering is an edge subset $A$ of $E$ such that $G-A$ has no $k-$clique. The minimum weight of $k-$cl…
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.
For a graph $X$ without isolated vertices and without isolated edges, a product-irregular labelling $ω:E(X)\rightarrow \{1,2,\ldots,s\}$, first defined by Anholcer in 2009, is a labelling of the edges of $X$ such that for any two distinct vertices $u$ and $v$ of $X$ the product of labels of the edges incident with $u$ is different from the product of labels of the edges incident with $v$. The minimal $s$ for which there exist a product irregular labeling is called the product irregularity strength of $X$ and is denoted by $ps(X)$. Clique cover number of a graph is the minimum number of cliques…
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.
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.
Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\sum_{i=1}^n d(G,i) x^i$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. For two graphs $G$ and $H$, let $\mathcal{C} = \{C_1,C_2, \cdots, C_k\}$ be a clique cover of $G$ and $U\subseteq V(H)$. We consider clique cover product which denoted by $G^\mathcal{C} \star H^U$ and obtained from $G$ as follows: for each clique $C_i \in \mathcal{C}$, add a copy of the graph $H$ and join every vertex of $C_i$ to every vertex of $U$. We prove that the domination polynomial of clique cov…
Let $G$ be a graph and $t\ge 0$. The largest reduced neighborhood clique cover number of $G$, denoted by ${\hatβ}_t(G)$, is the largest, overall $t$-shallow minors $H$ of $G$, of the smallest number of cliques that can cover any closed neighborhood of a vertex in $H$. It is known that ${\hatβ}_t(G)\le s_t$, where $G$ is an incomparability graph and $s_t$ is the number of leaves in a largest $t-$shallow minor which is isomorphic to an induced star on $s_t$ leaves. In this paper we give an overview of the properties of ${\hatβ}_t(G)$ including the connections to the greatest reduced average dens…
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…
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 $k$.
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 fi…
A graph is normal if it admits a clique cover $\mathcal C$ and a stable set cover $\mathcal S$ such that each clique in $\mathcal C$ and each stable set in $\mathcal S$ have a vertex in common. The pair $(\mathcal{C,S})$ is a normal cover of the graph. We present the following extremal property of normal covers. For positive integers $c,s$, if a graph with $n$ vertices admits a normal cover with cliques of sizes at most $c$ and stable sets of sizes at most $s$, then $c+s\geq\log_2(n)$. For infinitely many $n$, we also give a construction of a graph with $n$ vertices that admits a normal cover …
Indeterminate strings have received considerable attention in the recent past; see for example Christodoulakis et al 2015 and Helling et al 2017. This attention is due to their applicability in bioinformatics, and to the natural correspondence with undirected graphs. One aspect of this correspondence is the fact that the minimal alphabet size of indeterminates representing any given undirected graph corresponds to the size of the minimal clique cover of this graph. This paper solves a related problem proposed in Helling et al 2017: compute $Θ_n(m)$, which is the size of the largest possible mi…
Let $G$ be a graph and $t\ge 0$. A new graph parameter termed the largest reduced neighborhood clique cover number of $G$, denoted by ${\hatβ}_t(G)$, is introduced. Specifically, ${\hatβ}_t(G)$ is the largest, overall $t$-shallow minors $H$ of $G$, of the smallest number of cliques that can cover any closed neighborhood of a vertex in $H$. We verify that ${\hatβ}_t(G)=1$ when $G$ is chordal, and, ${\hatβ}_t(G)\le s$, where $G$ is an incomparability graph that does not have a $t-$shallow minor which is isomorphic to an induced star on $s$ leaves. Moreover, general properties of ${\hatβ}_t(G)$ i…
For a clique cover $C$ in the undirected graph $G$, the {\it clique cover graph} of $C$ is the graph obtained by contracting the vertices of each clique in $C$ into a single vertex. The {\it clique cover width} of $G$, denoted by $CCW(G)$, is the minimum value of the bandwidth of all clique cover graphs in $G$. Any $G$ with $CCW(G)=1$ is known to be an incomparability graph, and hence is called, a {\it unit incomparability graph}. We introduced the {\it unit incomparability dimension of $G$}, denoted by$Udim(G)$, to be the smallest integer $d$ so that there are unit incomparability graphs $H_i…
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）
Let $n,k,b$ be integers with $1 \le k-1 \le b \le n$ and let $G_{n,k,b}$ be the graph whose vertices are the $k$-element subsets $X$ of $\{0,\dots,n\}$ with $\max(X)-\min(X) \le b$ and where two such vertices $X,Y$ are joined by an edge if $\max(X \cup Y) - \min(X \cup Y) \le b$. These graphs are generated by applying a transformation to maximal $k$-uniform hypergraphs of bandwidth $b$ that is used to reduce the (weak) edge clique covering problem to a vertex clique covering problem. The bandwidth of $G_{n,k,b}$ is thus the largest possible bandwidth of any transformed $k$-uniform hypergraph o…
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…
A graph is said to be well-covered if all its maximal independent sets are of the same size. In 1999, Yamashita and Kameda introduced a subclass of well-covered graphs, called localizable graphs and defined as graphs having a partition of the vertex set into strong cliques, where a clique in a graph is strong if it intersects all maximal independent sets. Yamashita and Kameda observed that all well-covered trees are localizable, pointed out that the converse inclusion fails in general, and asked for a characterization of localizable graphs. In this paper we obtain several structural and algori…
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…
In this paper, we relate the competition number of a graph to its edge clique cover number by presenting a tight inequality $k(G) \ge θ_e(G)-|V(G)|+\widetilde{k}(G)$ where $θ_e(G)$, $k(G)$, and $\widetilde{k}(G)$ are the edge clique cover number, the competition number, and the co-competition number of a graph $G$, respectively. By utilizing this inequality and a notion of competition-effective edge clique cover, we obtain some meaningful results on competition numbers of planar graphs.
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…
A graph is $(H_1,H_2)$-free for a pair of graphs $H_1,H_2$ if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. In 2001, Král', Kratochv\'ıl, Tuza, and Woeginger initiated a study into the complexity of Colouring for $(H_1,H_2)$-free graphs. Since then, others have tried to complete their study, but many cases remain open. We focus on those $(H_1,H_2)$-free graphs where $H_2$ is $\overline{H_1}$, the complement of $H_1$. As these classes are closed under complementation, the computational complexities of Colouring and Clique Cover coincide. By combining new and known results, we ar…
A graph is called $P_t$-free} if it does not contain a $t$-vertex path as an induced subgraph. While $P_4$-free graphs are exactly cographs, the structure of $P_t$-free graphs for $t \geq 5$ remains little understood. On one hand, classic computational problems such as Maximum Weight Independent Set (MWIS) and $3$-Coloring are not known to be NP-hard on $P_t$-free graphs for any fixed $t$. On the other hand, despite significant effort, polynomial-time algorithms for MWIS in $P_6$-free graphs~[SODA 2019] and $3$-Coloring in $P_7$-free graphs~[Combinatorica 2018] have been found only recently. I…
A complete graph is the graph in which every two vertices are adjacent. For a graph $G=(V,E)$, the complete width of $G$ is the minimum $k$ such that there exist $k$ independent sets $\mathtt{N}_i\subseteq V$, $1\le i\le k$, such that the graph $G'$ obtained from $G$ by adding some new edges between certain vertices inside the sets $\mathtt{N}_i$, $1\le i\le k$, is a complete graph. The complete width problem is to decide whether the complete width of a given graph is at most $k$ or not. In this paper we study the complete width problem. We show that the complete width problem is NP-complete o…
