A solitaire game played on 2-colored graphs

We introduce a solitaire game played on a graph. Initially one disk is placed at each vertex, one face green and the other red, oriented with either color facing up. Each move of the game consists of selecting a vertex whose disk shows green, flipping over the disks at neighboring vertices, and deleting the selected vertex. The game is won if all vertices are eliminated. We derive a simple parity-based necessary condition for winnability of a given game instance. By studying graph operations that construct new graphs from old ones, we obtain broad classes of graphs where this condition also suffices, thus characterizing the winnable games on such graphs. Concerning two familiar (but narrow) classes of graphs, we show that for trees a game is winnable if and only if the number of green vertices is odd, and for n-cubes a game is winnable if and only if the number of green vertices is even and not all vertices have the same color. We provide a linear-time algorithm for deciding winnability for games on maximal outerplanar graphs. We reduce the decision problem for winnability of a game on an arbitrary graph G to winnability of games on its blocks, and to winnability on homeomorphic images of G obtained by contracting edges at 2-valent vertices.     

A note on triangulating the 5-cube

We show that any triangulation of the 5-cube I⁵ by complete truncation, i.e., ‘slicing off’ the even (or the odd) vertices, cannot use less than 67 or more than 68 pieces.     

Integral trees of odd diameters

A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. Recently, Csikvári proved the existence of integral trees of any even diameter. In the odd case, integral trees have been constructed with diameter at most 7. In this article, we show that for every odd integer n>1, there are infinitely many integral trees of diameter n. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Symmetry in de Branges almost Pontryagin spaces

In many examples of de Branges spaces symmetry appears naturally. Presence of symmetry gives rise to a decomposition of the space into two parts, the ‘even’ and the ‘odd’ part, which themselves can be regarded as de Branges spaces. The converse question is to decide whether a given space is the ‘even’ part or the ‘odd’ part of some symmetric space, and, if yes, to describe the totality of all such symmetric spaces. We consider this question in an indefinite (almost Pontryagin space) setting, and give a complete answer. Interestingly, it turns out that the answers for the ‘even’ and ‘odd’ cases read quite differently; the latter is significantly more complex.     

A proof of Hougardy''s conjecture for diamond-free graphs

A pair of vertices of a graph is called an even pair if every chordless path between them has an even number of edges. A graph is minimally even pair free if it is not a clique, contains no even pair, but every proper induced subgraph either contains an even pair or is a clique. Hougardy (European J. Combin. 16 (1995) 17–21) conjectured that a minimally even pair free graph is either an odd cycle of length at least five, the complement of an even or odd cycle of length at least five, or the linegraph of a bipartite graph. A diamond is a graph obtained from a complete graph on four vertices by removing an edge. In this paper we verify Hougardy's conjecture for diamond-free graphs by adapting the characterization of perfect diamond-free graphs given by Fonlupt and Zemirline (Maghreb Math. Rev. 1 (1992) 167–202).     

Cycles through specified vertices of a graph

We prove that ifS is a set ofk−1 vertices in ak-connected graphG, then the cycles throughS generate the cycle space ofG. Moreover, whenk≧3, each cycle ofG can be expressed as the sum of an odd number of cycles throughS. On the other hand, ifS is a set ofk vertices, these conclusions do not necessarily hold, and we characterize the exceptional cases. As corollaries, we establish the existence of odd and even cycles through specified vertices and deduce the existence of long odd and even cycles in graphs of high connectivity.     

一类优美树

§1. Preliminary Lemma Using A to denote the cardinal number of a finite set A, we do not explain anymore what are similar to A in this paper. For a given tree T(V,E), if there is a labeling f of its vertices, which satisfiesf［V(T)］={f(u)u∈V(T)}={0,1,2,…,E(T)},lf［E(T)］={lf(uv)=f(u)-f(v), uv∈E(T)}={1,2,…,E(T)},then f is called a graceful labeling of T, and T is called a graceful tree. lf denotes the labeling of edges that is derived from f. In the 1960s, RingedKotzing and A. R…     

Expanding graphs contain all small trees

The assertion of the title is formulated and proved. The result is then used to construct graphs with a linear number of edges that, even after the deletion of almost all of their edges or almost all of their vertices, continue to contain all small trees.     

A greedy algorithm for multicut and integral multiflow in rooted trees

We present an O(min(Kn,n²)) algorithm to solve the maximum integral multiflow and minimum multicut problems in rooted trees, where K is the number of commodities and n is the number of vertices. These problems are NP-hard in undirected trees but polynomial in directed trees. In the algorithm we propose, we first use a greedy procedure to build the multiflow then we use duality properties to obtain the multicut and prove the optimality.     

Combinatorics of rooted trees and Hopf algebras

We begin by considering the graded vector space with a basis consisting of rooted trees, with grading given by the count of non-root vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of vertices, and each operator naturally associates a multiplicity to each pair of rooted trees. By using symmetry groups of trees we define an inner product with respect to which the growth and pruning operators are adjoint, and obtain several results about the associated multiplicities.

Now the symmetric algebra on the vector space of rooted trees (after a degree shift) can be endowed with a coproduct to make a Hopf algebra; this was defined by Kreimer in connection with renormalization. We extend the growth and pruning operators, as well as the inner product mentioned above, to Kreimer's Hopf algebra. On the other hand, the vector space of rooted trees itself can be given a noncommutative multiplication: with an appropriate coproduct, this leads to the Hopf algebra of Grossman and Larson. We show that the inner product on rooted trees leads to an isomorphism of the Grossman-Larson Hopf algebra with the graded dual of Kreimer's Hopf algebra, correcting an earlier result of Panaite.

     

On the Laplacian spectral radius of weighted trees with a positive weight set

The spectrum of weighted graphs is often used to solve the problems in the design of networks and electronic circuits. We first give some perturbational results on the (signless) Laplacian spectral radius of weighted graphs when some weights of edges are modified; we then determine the weighted tree with the largest Laplacian spectral radius in the set of all weighted trees with a fixed number of pendant vertices and a positive weight set. Furthermore, we also derive the weighted trees with the largest Laplacian spectral radius in the set of all weighted trees with a fixed positive weight set and independence number, matching number or total independence number.     

亏量为一的树的三个结论

一个图的亏量是指不能被某个最大匹配所覆盖的顶点数.本文通过三个结论刻画了亏量为一的树.     

2-noncrossing trees and 5-ary trees

Recently, Gu et al. [N.S.S. Gu, N.Y. Li, T. Mansour, 2-Binary trees: Bijections and related issues, Discrete Math. 308 (2008) 1209-1221] introduced 2-binary trees and 2-plane trees which are closely related to ternary trees. In this note, we study the 2-noncrossing tree, a noncrossing tree in which each vertex is colored black or white and there is no ascent (u,v) such that both the vertices u and v are colored black. By using the representation of Panholzer and Prodinger for noncrossing trees, we find a correspondence between the set of 2-noncrossing trees of n edges with a black root and the set of 5-ary trees with n internal vertices.     

Maximum Distance Between Slater Orders and Copeland Orders of Tournaments

Given a tournament T?=?(X, A), we consider two tournament solutions applied to T: Slater’s solution and Copeland’s solution. Slater’s solution consists in determining the linear orders obtained by reversing a minimum number of directed edges of T in order to make T transitive. Copeland’s solution applied to T ranks the vertices of T according to their decreasing out-degrees. The aim of this paper is to compare the results provided by these two methods: to which extent can they lead to different orders? We consider three cases: T is any tournament, T is strongly connected, T has only one Slater order. For each one of these three cases, we specify the maximum of the symmetric difference distance between Slater orders and Copeland orders. More precisely, thanks to a result dealing with arc-disjoint circuits in circular tournaments, we show that this maximum is equal to n(n???1)/2 if T is any tournament on an odd number n of vertices, to (n ²???3n?+?2)/2 if T is any tournament on an even number n of vertices, to n(n???1)/2 if T is strongly connected with an odd number n of vertices, to (n ²???3n???2)/2 if T is strongly connected with an even number n of vertices greater than or equal to 8, to (n ²???5n?+?6)/2 if T has an odd number n of vertices and only one Slater order, to (n ²???5n?+?8)/2 if T has an even number n of vertices and only one Slater order.     

Remarks on the critical graph conjecture

The vertex-critical graph conjecture (critical graph conjecture respectively) states that every vertex-critical (critical) graph has an odd number of vertices. In this note we prove that if G is a critical graph of even order, then G has at least three vertices of less-than-maximum valency. In addition, if G is a 3-critical multigraph of smallest even order, then G is simple and has no triangles.     

Maximizing and minimizing the number of generalized colorings of trees

We classify the trees on $n$ vertices with the maximum and the minimum number of certain generalized colorings, including conflict-free, odd, non-monochromatic, star, and star rainbow vertex colorings. We also extend a result of Cutler and Radcliffe on the maximum and minimum number of existence homomorphisms from a tree to a completely looped graph on $q$ vertices.     

On defect-d matchings in graphs

A defect-d matching in a graph G is a matching which covers all but d vertices of G. G is d-covered if each edge of G belongs to a defect-d matching. Here we characterise d-covered graphs and d-covered connected bipartite graphs. We show that a regular graph G of degree r which is (r ? 1)-edge-connected is 0-covered or 1-covered depending on whether G has an even or odd number of vertices, but, given any non-negative integers r and d, there exists a graph regular of degree r with connectivity and edge-connectivity r ? 2 which does not even have a defect-d matching. Finally, we prove that a vertex-transitive graph is 0-covered or 1-covered depending on whether it has an even or odd number of vertices.     

递归树的若干枚举特征

递归树由Meir和Moon定义作非平面增长树的一种，且所有节点出度都是允许的．本文首先在n个节点的递归树集合和n-1个元素的排列之间建立一个新的──对应，这个对应能同时给出树叶子和排列中的路段之间的对应和树叶子数和排列中的路段数之间的密切关系．同时还研究递归树的各种枚举特征，诸如节点的分类枚举（内节点和叶子节点、偶节点和奇节点，具不同出度的节点）和通路长度枚举（接各种节点分类）．