Graphs considered in this paper are finite, undirected and without loops, but with multiple edges. For an integer $t\geq 1$, denote by $\mathcal{MG}_t$ the class of graphs whose maximum multiplicity is at most $t$. A graph $G$ is called strictly $t$-degenerate if every non-empty subgraph $H$ of $G$ contains a vertex $v$ whose degree in $H$ is at most $t-1$. The point partition number $χ_t(G)$ of $G$ is smallest number of colors needed to color the vertices of $G$ so that each vertex receives a color and vertices with the same color induce a strictly $t$-degenerate subgraph of $G$. So $χ_1$ is the chromatic number, and $χ_2$ is known as the point aboricity. The point partition number $χ_t$ with $t\geq 1$ was introduced by Lick and White. If $H$ is a simple graph, then $tH$ denotes the graph obtained from $H$ by replacing each edge of $H$ by $t$ parallel edges. Then $ω_t(G)$ is the largest integer $n$ such that $G$ contains a $tK_n$ as a subgraph. Let $G$ be a graph belonging to $\mathcal{MG}_t$. Then $ω_t(G)\leq χ_t(G)$ and we say that $G$ is $χ_t$-perfect if every induced subgraph $H$ of $G$ satisfies $ω_t(H)=χ_t(H)$. Based on the Strong Perfect Graph Theorem due to Chudnowsky, Robertson, Seymour and Thomas, we give a characterization of $χ_t$-perfect graphs of $\mathcal{MG}_t$ by a set of forbidden induced subgraphs. We also discuss some complexity problems for the class of $χ_t$-critical graphs.
A graph has a perfect partition if all its perfect matchings can be partitioned so that each part is a 1-factorization of the graph. Let $L_{rm, r}=K_{rm,rm}-mK_{r,r}$. We first give a formula to count the number of perfect matchings of $L_{rm, r}$, then show that $L_{6,1}$ and $L_{8,2}$ have perfect partitions.
Let $X_1,\dots, X_n$ be independent integers distributed uniformly on $\{1,\dots, M\}$, $M=M(n)\to\infty$ however slow. A partition $S$ of $[n]$ into $ν$ non-empty subsets $S_1,\dots, S_ν$ is called perfect, if all $ν$ values $\sum_{j\in S_{\a}}X_j$ are equal. For a perfect partition to exist, $\sum_j X_j$ has to be divisible by $ν$. For $ν=2$, Borgs et al. proved, among other results, that, conditioned on $\sum_j X_j$ being even, with high probability a perfect partition exists if $κ:=\lim \tfrac{n}{\log M}>\tfrac{1}{\log 2}$, and that w.h.p. no perfect partition exists if $κ<\tfrac{1}{\log 2…
We find Stieltjes-type and Jacobi-type continued fractions for some "master polynomials" that enumerate permutations, set partitions or perfect matchings with a large (sometimes infinite) number of simultaneous statistics. Our results contain many previously obtained identities as special cases, providing a common refinement of all of them.
This paper is concerned with structures of general graphs with perfect matchings. We first reveal a partially ordered structure among factor-components of general graphs with perfect matchings. Our second result is a generalization of Kotzig's canonical partition to a decomposition of general graphs with perfect matchings. It contains a short proof for the theorem of the canonical partition. These results give decompositions which are canonical, that is, unique to given graphs. We also show that there are correlations between these two and that these can be computed in polynomial time.
Let $H$ be a $3$-partite $3$-uniform hypergraph, i.e. a $3$-uniform hypergraph such that every edge intersects every partition class in exactly one vertex, with each partition class of size $n$. We determine a Dirac-type vertex degree threshold for perfect matchings in $3$-partite $3$-uniform hypergraphs.
We prove that every $1$-error-correcting code over a finite field can be embedded in a $1$-perfect code of some larger length. Embedding in this context means that the original code is a subcode of the resulting $1$-perfect code and can be obtained from it by repeated shortening. Further, we generalize the results to partitions: every partition of the Hamming space into $1$-error-correcting codes can be embedded in a partition of a space of some larger dimension into $1$-perfect codes. For the partitions, the embedding length is close to the theoretical bound for the general case and optimal f…
A graph $G$ is said to be $2$-divisible if for all (nonempty) induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A,B$ such that $ω(A) < ω(H)$ and $ω(B) < ω(H)$. A graph $G$ is said to be perfectly divisible if for all induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A,B$ such that $H[A]$ is perfect and $ω(B) < ω(H)$. We prove that if a graph is $(P_5,C_5)$-free, then it is $2$-divisible. We also prove that if a graph is bull-free and either odd-hole-free or $P_5$-free, then it is perfectly divisible.
We show there is an uncountable number of parallel total perfect codes in the integer lattice graph $Λ$ of $\R^2$. In contrast, there is just one 1-perfect code in $Λ$ and one total perfect code in $Λ$ restricting to total perfect codes of rectangular grid graphs (yielding an asymmetric, Penrose, tiling of the plane). We characterize all cycle products $C_m\times C_n$ with parallel total perfect codes, and the $d$-perfect and total perfect code partitions of $Λ$ and $C_m\times C_n$, the former having as quotient graph the undirected Cayley graphs of $\Z_{2d^2+2d+1}$ with generator set $\{1,2d^…
In this paper, we deal with a simple geometric problem: Is it possible to partition a rectangle into $k$ non-congruent rectangles of equal area? This problem is motivated by the so-called `Mondrian art problem' that asks a similar question for dissections with rectangles of integer sides. Here, we generalize the Mondrian problem by allowing rectangles of real sides. In this case, we show that the minimum value of $k$ for a rectangle to have a `perfect Mondrian partition' (that is, with non-congruent equal-area rectangles) is seven. Moreover, we prove that such a partition is unique (up to symm…
The Mondrian problem consists of dissecting a square of side length $n\in \NN$ into non-congruent rectangles with natural length sides such that the difference $d(n)$ between the largest and the smallest areas of the rectangles partitioning the square is minimum. In this paper, we compute some bounds on $d(n)$ in terms of the number of rectangles of the square partition. These bounds provide us optimal partitions for some values of $n \in \NN$. We provide a sequence of square partitions such that $d(n)/n^2$ tends to zero for $n$ large enough. For the case of `perfect' partitions, that is, with…
We obtain limit shape of perfect matchings on a large class of rail-yard graphs with right boundary condition given by the empty partition, and left boundary condition given by either by a staircase partition with constant density or a piecewise partition with densities either 1 or 0. We prove the parametric equations for the frozen boundary, and find conditions under which the frozen boundary is a cloud curve, or a union of disjoint cloud curves.
The partition of graphs into "nice" subgraphs is a central algorithmic problem with strong ties to matching theory. We study the partitioning of undirected graphs into same-size stars, a problem known to be NP-complete even for the case of stars on three vertices. We perform a thorough computational complexity study of the problem on subclasses of perfect graphs and identify several polynomial-time solvable cases, for example, on interval graphs and bipartite permutation graphs, and also NP-complete cases, for example, on grid graphs and chordal graphs.
In a recent work, Keith and Xiong gave a refinement of Glaisher's theorem by using a Sylvester-style bijection. In this paper, we introduce two families of colored partitions, flat and regular partitions, and generalize the bijection of Keith and Xiong to these partitions. We then state two results, the first at degree one, where partitions have parts with primary colors, and the second result at degree two for secondary-colored partitions, using the result of the first paper of this series. These results allow us to easily retrieve the Frenkel-Kac character formulas of level one standard modu…
Let $H$ be an induced subgraph of the torus $C_k^m$. We show that when $k \ge 3$ is even and $|V(H)|$ divides some power of $k$, then for sufficiently large $n$ the torus $C_k^n$ has a perfect vertex-packing with induced copies of $H$. On the other hand, disproving a conjecture of Gruslys, we show that when $k$ is odd and not a prime power, then there exists $H$ such that $|V(H)|$ divides some power of $k$, but there is no $n$ such that $C_k^n$ has a perfect vertex-packing with copies of $H$. We also disprove a conjecture of Gruslys, Leader and Tan by exhibiting a subgraph $H$ of the $k$-dimen…
In this paper we obtain the necessary condition for the existence of perfect $k$-colorings (equitable $k$-partitions) in Hamming graphs $H(n,q)$, where $q=2,3,4$ and Doob graphs $D(m,n)$. As an application, we prove the non-existence of extended perfect codes in $H(n,q)$, where $q=3,4$, $n>q+2$, and in $D(m,n)$, where $2m+n>6$.
Recently, Rattan and the first author (Ann. Comb. 25 (2021) 697-728) proved a conjectured inequality of Berkovich and Uncu (Ann. Comb. 23 (2019) 263-284) concerning partitions with an impermissible part. In this article, we generalize this inequality upon considering t impermissible parts. We compare these with partitions whose certain parts appear with a frequency which is a perfect t^{th} power. Our inequalities hold after a certain bound, which for given t is a polynomial in s, a major improvement over the previously known bound in the case t=1. To prove these inequalities, our methods invo…
We describe new constructions of graphs which exhibit perfect state transfer on continuous-time quantum walks. Our constructions are based on variants of the double cones [BCMS09,ANOPRT10,ANOPRT09] and the Cartesian graph products (which includes the n-cube) [CDDEKL05]. Some of our results include: (1) If $G$ is a graph with perfect state transfer at time $t_{G}$, where $t_{G}\Spec(G) \subseteq \ZZπ$, and $H$ is a circulant with odd eigenvalues, their weak product $G \times H$ has perfect state transfer. Also, if $H$ is a regular graph with perfect state transfer at time $t_{H}$ and $G$ is a g…
A graph is {\em perfect} if, in all its induced subgraphs, the size of a largest clique is equal to the chromatic number. Examples of perfect graphs include bipartite graphs, line graphs of bipartite graphs and the complements of such graphs. These four classes of perfect graphs will be called {\em basic}. In 1960, Berge formulated two conjectures about perfect graphs, one stronger than the other. The weak perfect graph conjecture, which states that a graph is perfect if and only if its complement is perfect, was proved in 1972 by Lovász. This result is now known as the perfect graph theorem. …
We revisit the following problem (along with its higher dimensional variant): Given a set $S$ of $n$ points inside an axis-parallel rectangle $U$ in the plane, find a maximum-area axis-parallel sub-rectangle that is contained in $U$ but contains no points of $S$. (I) We present an algorithm that finds a large empty box amidst $n$ points in $[0,1]^d$: a box whose volume is at least $\frac{\log{d}}{4(n + \log{d})}$ can be computed in $O(n+d \log{d})$ time. (II) To better analyze the above approach, we introduce the concepts of perfect vector sets and properly overlapping partitions, in connectio…
We define a perfect coloring of a graph $G$ as a proper coloring of $G$ such that every connected induced subgraph $H$ of $G$ uses exactly $ω(H)$ many colors where $ω(H)$ is the clique number of $H$. A graph is perfectly colorable if it admits a perfect coloring. We show that the class of perfectly colorable graphs is exactly the class of perfect paw-free graphs. It follows that perfectly colorable graphs can be recognized and colored in linear time.
We show that the following problems are NP-complete. 1. Can the vertex set of a graph be partitioned into two sets such that each set induces a perfect graph? 2. Is the difference between the chromatic number and clique number at most $1$ for every induced subgraph of a graph? 3. Can the vertex set of every induced subgraph of a graph be partitioned into two sets such that the first set induces a perfect graph, and the clique number of the graph induced by the second set is smaller than that of the original induced subgraph? 4. Does a graph contain a stable set whose deletion results in a perf…
A graph is strongly perfect if every induced subgraph H has a stable set that meets every nonempty maximal clique of H. The characterization of strongly perfect graphs by a set of forbidden induced subgraphs is not known. Here we provide several new minimal non-strongly-perfect graphs.
Given non-negative weights w_S on the k-subsets S of a km-element set V, we consider the sum of the products w_{S_1} ... w_{S_m} for all partitions V = S_1 cup ... cup S_m into pairwise disjoint k-subsets S_i. When the weights w_S are positive and within a constant factor, fixed in advance, of each other, we present a simple polynomial time algorithm to approximate the sum within a polynomial in m factor. In the process, we obtain higher-dimensional versions of the van der Waerden and Bregman-Minc bounds for permanents. We also discuss applications to counting of perfect and nearly perfect mat…
Inspired by a question of Yannakakis on the Vertex Packing polytope of perfect graphs, we study the Clique-Stable Set Separation in a non-hereditary subclass of perfect graphs. A cut (B,W) of G (a bipartition of V(G)) separates a clique K and a stable set S if $K\subseteq B$ and $S\subseteq W$. A Clique-Stable Set Separator is a family of cuts such that for every clique K, and for every stable set S disjoint from K, there exists a cut in the family that separates K and S. Given a class of graphs, the question is to know whether every graph of the class admits a Clique-Stable Set Separator cont…
The subject of this contribution is the study of the Lovász-Schrijver PSD-operator N+ applied to the edge relaxation of the stable set polytope of a graph. We are particularly interested in the problem of characterizing graphs for which N+ generates the stable set polytope in one step, called N+-perfect graphs. It is conjectured that the only N+-perfect graphs are those whose stable set polytope is described by inequalities with near-bipartite support. So far, this conjecture has been proved for near-perfect graphs, fs-perfect graphs, and webs. Here, we verify it for line graphs, by proving th…
We study the linear extension complexity of stable set polytopes of perfect graphs. We make use of known structural results permitting to decompose perfect graphs into basic perfect graphs by means of two graph operations: 2-join and skew partitions. Exploiting the link between extension complexity and the nonnegative rank of an associated slack matrix, we investigate the behaviour of the extension complexity under these graph operations. We show bounds for the extension complexity of the stable set polytope of a perfect graph $G$ depending linearly on the size of $G$ and involving the depth o…
In Partition Into Complementary Subgraphs (Comp-Sub) we are given a graph $G=(V,E)$, and an edge set property $Π$, and asked whether $G$ can be decomposed into two graphs, $H$ and its complement $\overline{H}$, for some graph $H$, in such a way that the edge cut $[V(H),V(\overline{H})]$ satisfies the property $Π$. Motivated by previous work, we consider Comp-Sub($Π$) when the property $Π=\mathcal{PM}$ specifies that the edge cut of the decomposition is a perfect matching. We prove that Comp-Sub($\mathcal{PM}$) is GI-hard when the graph $G$ is $\{C_{k\geq 7}, \overline{C}_{k\geq 7} \}$-free. On…
We study a family of graphs with diameter two and asymptotically optimal order for their maximum degree, obtained from perfect difference sets. We show that for all known examples of perfect difference sets, the graph we obtain is isomorphic to one of the Brown graphs, a well-known family of graphs in the degree-diameter problem.
Let $γ_g(G)$ and $γ_{tg}(G)$ be the game domination number and the total game domination number of a graph $G$, respectively. Then $G$ is $γ_g$-perfect (resp. $γ_{tg}$-perfect), if every induced subgraph $F$ of $G$ satisfies $γ_g(F)=γ(F)$ (resp. $γ_{tg}(F)=γ_t(F)$). A recursive characterization of $γ_g$-perfect graphs is derived. The characterization yields a polynomial recognition algorithm for $γ_g$-perfect graphs. It is proved that every minimally $γ_g$-imperfect graph has domination number $2$. All minimally $γ_g$-imperfect triangle-free graphs are determined. It is also proved that $γ_{tg…
We give exact formulas for the transmission (i.e. the sum of all distances between vertices) of perfect trees and rooted powers of (connected finite) graphs.
It has been conjectured that for every claw-free graph $G$ the choice number of $G$ is equal to its chromatic number. We focus on the special case of this conjecture where $G$ is perfect. Claw-free perfect graphs can be decomposed via clique-cutset into two special classes called elementary graphs and peculiar graphs. Based on this decomposition we prove that the conjecture holds true for every claw-free perfect graph with maximum clique size at most $4$.
In this paper, we study the problem that which of distance-regular graphs admit a perfect $1$-code. Among other results, we characterize distance-regular line graphs which admit a perfect $1$-code. Moreover, we characterize all known distance-regular graphs with small valency at most $4$, the distance-regular graphs with known putative intersection arrays for valency $5$, and all distance-regular graphs with girth $3$ and valency $6$ or $7$ which admit a perfect $1$-code.
We show that each perfect matching in a bipartite graph $G$ intersects at least half of the perfect matchings in $G$. This result has equivalent formulations in terms of the permanent of the adjacency matrix of a graph, and in terms of derangements and permutations on graphs. We give several related results and open questions.
Let $γ(G)$ and $β(G)$ denote the domination number and the covering number of a graph $G$, respectively. A connected non-trivial graph $G$ is said to be $γβ$-{perfect} if $γ(H)=β(H)$ for every non-trivial induced connected subgraph $H$ of $G$. In this note we present an elementary proof of a characterization of the $γβ$-perfect graphs.
Inspired by a famous characterization of perfect graphs due to Lovász, we define a graph $G$ to be sum-perfect if for every induced subgraph $H$ of $G$, $α(H) + ω(H) \geq |V(H)|$. (Here $α$ and $ω$ denote the stability number and clique number, respectively.) We give a set of $27$ graphs and we prove that a graph $G$ is sum-perfect if and only if $G$ does not contain any of the graphs in the set as an induced subgraph.
The middle graph $M(G)$ of a graph $G$ is the graph obtained by subdividing each edge of $G$ exactly once and joining all these newly introduced vertices of adjacent edges of $G$. A perfect Roman dominating function on a graph $G$ is a function $f : V(G) \rightarrow \{0, 1, 2\}$ satisfying the condition that every vertex $v$ with $f(v)=0$ is adjacent to exactly one vertex $u$ for which $f(u)=2$. The weight of a perfect Roman dominating function $f$ is the sum of weights of vertices. The perfect Roman domination number is the minimum weight of a perfect Roman dominating function on $G$. In this…
In this paper we investigate invertibility of graphs with a unique perfect matching, i.e. graphs having a unique 1-factor. We recall the new notion of the so-called negatively invertible graphs investigated by the authors in the recent paper. It is an extension of the classical definition of an inverse graph due to Godsil. We characterize all graphs with a unique perfect matching on $m\le 6$ vertices with respect to their positive and negative invertibility. We show that negatively invertible graphs exhibit properties like selfinvertibility which cannot be observed within the class of positive…
We narrow in on the number of graphical partitions for which there is no known generating function by manipulating the well known generating function for Frobenius partitions.
Perfect matching in 4-partite 4-uniform hypergraphs
School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；School of Mathematical Sciences, CMA-Shanghai, Shanghai Jiao Tong University（上海交通大学数学科学学院，上海数学中心）
Let $G=(V,E)$ be a graph and let $A_G$ be the clique-vertex incidence matrix of $G$. It is well known that $G$ is perfect iff the system $A_{_G}\mathbf x\le \mathbf 1$, $\mathbf x\ge\mathbf0$ is totally dual integral (TDI). In 1982, Cameron and Edmonds proposed to call $G$ box-perfect if the system $A_{_G}\mathbf x\le \mathbf 1$, $\mathbf x\ge\mathbf0$ is box-totally dual integral (box-TDI), and posed the problem of characterizing such graphs. In this paper we prove the Cameron-Edmonds conjecture on box-perfectness of parity graphs, and identify several other classes of box-perfect graphs. We …
