平衡子欧拉符号图及符号线图

符号图$S=(S^u,\sigma)$是以$S^u$作为底图并且满足$\sigma: E(S^u)\rightarrow\{+,-\}$. 设$E^-(S)$表示$S$的负边集. 如果$S^u$是欧拉的(或者分别是子欧拉的, 欧拉的且$|E^-(S)|$是偶数, 则$S$是欧拉符号图(或者分别是子欧拉符号图, 平衡欧拉符号图). 如果存在平衡欧拉符号图$S''$使得$S''$由$S$生成, 则$S$是平衡子欧拉符号图. 符号图$S$的线图$L(S)$也是一个符号图, 使得$L(S)$的点是$S$中的边, 其中$e_ie_j$是$L(S)$中的边当且仅当$e_i$和$e_j$在$S$中相邻,并且$e_ie_j$是$L(S)$中的负边当且仅当$e_i$和$e_j$在$S$中都是负边. 本文给出了两个符号图族$S$和$S''$,它们应用于刻画平衡子欧拉符号图和平衡子欧拉符号线图. 特别地, 本文证明了符号图$S$是平衡子欧拉的当且仅当$\not\in S$, $S$的符号线图是平衡子欧拉的当且仅当$S\not\in S''$.     

On the Characterization of Maximal Planar Graphs with a Given Signed Cycle Domination Number

Let G =(V, E) be a simple graph. A function f : E → {+1,-1} is called a signed cycle domination function(SCDF) of G if ∑_(e∈E(C))f(e) ≥ 1 for every induced cycle C of G. The signed cycle domination number of G is defined as γ'_(sc)(G) = min{∑_(e∈E)f(e)| f is an SCDF of G}. This paper will characterize all maximal planar graphs G with order n ≥ 6 and γ'_(sc)(G) = n.     

Balanced group-labeled graphs

A group-labeled graph is a graph whose vertices and edges have been assigned labels from some abelian group. The weight of a subgraph of a group-labeled graph is the sum of the labels of the vertices and edges in the subgraph. A group-labeled graph is said to be balanced if the weight of every cycle in the graph is zero. We give a characterization of balanced group-labeled graphs that generalizes the known characterizations of balanced signed graphs and consistent marked graphs. We count the number of distinct balanced labellings of a graph by a finite abelian group $\Gamma$ and show that this number depends only on the order of $\Gamma$ and not its structure. We show that all balanced labellings of a graph can be obtained from the all-zero labeling using simple operations. Finally, we give a new constructive characterization of consistent marked graphs and markable graphs, that is, graphs that have a consistent marking with at least one negative vertex.     

Intrinsically linked signed graphs in projective space

We define a signed embedding of a signed graph into real projective space to be an embedding such that an embedded cycle is 0-homologous if and only if it is balanced. We characterize signed graphs that have a linkless signed embedding. In particular, we exhibit 46 graphs that form the complete minor-minimal set of signed graphs that contain a non-split link for every signed embedding. With one trivial exception, these graphs are derived from different signings of the seven Petersen family graphs.     

Random packing by ρ-connected ρ-regular graphs

A graph G is packable by the graph F if its edges can be partitioned into copies of F. If deleting the edges of any F-packable subgraph from G leaves an F-packable graph, then G is randomly F-packable. If G is F-packable but not randomly F-packable then G is F-forbidden. The minimal F-forbidden graphs provide a characterization of randomly F-packable graphs. We show that for each ρ-connected ρ-regular graph F with ρ > 1, there is a set (F) of minimal F-forbidden graphs of a simple form, such that any other minimal F-forbidden graph can be obtained from a graph in (F) by a process of identifying vertices and removing copies of F. When F is a connected strongly edge-transitive graph having more than one edge (such as a cycle or hypercube), there is only one graph in (F).     

Conditions for the positivity of determinants

Consider a matrix with positive diagonal entries, which is similar via a positive diagonal matrix to a symmetric matrix, and whose signed directed graph has the property that if a cycle and its symmetrically placed complement have the same sign, then they are both positive. We provide sufficient conditions so that A be a P-matrix, that is , a matrix whose principal minors are all positive. We further provide sufficiet conditions for an arbitrary matrix A whose (undirected) graph is subordinate to a tree, to be a P-matrix. If, in additionA is sign symmetric and its undirected graph is a tree, we obtain necessary and sufficient conditions that it be a P-matrix. We go on to consider the positive semi-definiteness of symmetric matrices whose graphs are subordinate to a given tree and discuss the convexity of the set of all such matrices.     

Vertex pancyclic graphs

Let G be a graph of order n. A graph G is called pancyclic if it contains a cycle of length k for every 3kn, and it is called vertex pancyclic if every vertex is contained in a cycle of length k for every 3kn. In this paper, we shall present different sufficient conditions for graphs to be vertex pancyclic.     

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).     

Enumeration of weak isomorphism classes of signed graphs

A signed graph is a graph in which each line has a plus or minus sign. Two signed graphs are said to be weakly isomorphic if their underlying graphs are isomorphic through a mapping under which signs of cycles are preserved, the sign of a cycle being the product of the signs of its lines. Some enumeration problems implied by such a definition, including the problem of self-dual configurations, are solved here for complete signed graphs by methods of linear algebra over the two-element field. It is also shown that weak isomorphism classes of complete signed graphs are equal in number to other configurations: unlabeled even graphs, two-graphs and switching classes.     

On Homotopy of connected graphs having the same degree function

Let denote the set of simple graphs having the degree function . It is known that any two graphs from are homotopic in the sense that one can be obtained from the other by a series of simple two-edge switches so that every intermediate graph is in . Let denote the set of simple graphs having at most k components. A natural question arises whether it is possible to extend the above homotopy result on the class , and in particular, whether it is possible to obtain a connected simple graph from another connected simple graph of the same degree function by a series of two edge switches preserving connectivity. In this paper we give an answer to this question. We also consider two-edge switches of a special type and single out some classes of graphs for which these special operations provide the corresponding homotopy between its members.     

符号圈拼接图的零度

令$\eta(\Gamma)$和$c(\Gamma)$是符号图$\Gamma$的零度和基本圈数. 一个符号圈拼接图是指每个块都是圈的连通符号图. 本文证明了对任意符号拼接图$\eta(\Gamma)\le c(\Gamma)+1$成立, 并且刻画了等号成立的极图, 推广了王登银等人(2022)在简单圈拼接图上的结果. 此外, 我们证明了任意的符号拼接图$\eta(\Gamma)\neq c(\Gamma)$, 给出了满足$\eta(\Gamma)=c(\Gamma)-1$的符号拼接图的一些性质并刻画处$\eta(\Gamma)=c(\Gamma)-1$的二部符号拼接图.     

A note on compact graphs

An undirected simple graph G is called compact iff its adjacency matrix A is such that the polytope S(A) of doubly stochastic matrices X which commute with A has integral-valued extremal points only. We show that the isomorphism problem for compact graphs is polynomial. Furthermore, we prove that if a graph G is compact, then a certain naive polynomial heuristic applied to G and any partner G′ decides correctly whether G and G′ are isomorphic or not. In the last section we discuss some compactness preserving operations on graphs.     

The reduction of graph families closed under contraction

Let be a family of graphs. Suppose there is a nontrivial graph H such that for any supergraph G of H, G is in if and only if the contraction G/H is in . Examples of such an : graphs with a spanning closed trail; graphs with at least k edge-disjoint spanning trees; and k-edge-connected graphs (k fixed). We give a reduction method using contractions to find when a given graph is in and to study its structure if it is not in . This reduction method generalizes known special cases.     

Weakly triangulated graphs

A graph is triangulated if it has no chordless cycle with four or more vertices. It follows that the complement of a triangulated graph cannot contain a chordless cycle with five or more vertices. We introduce a class of graphs (namely, weakly triangulated graphs) which includes both triangulated graphs and complements of triangulated graphs (we define a graph as weakly triangulated if neither it nor its complement contains a chordless cycle with five or more vertices). Our main result is a structural theorem which leads to a proof that weakly triangulated graphs are perfect.     

Six-flows on almost balanced signed graphs

In 1983, Bouchet conjectured that every flow-admissible signed graph admits a nowhere-zero 6-flow. By Seymour's 6-flow theorem, Bouchet's conjecture holds for signed graphs with all edges positive. Recently, Rollová et al proved that every flow-admissible signed cubic graph with two negative edges admits a nowhere-zero 7-flow, and admits a nowhere-zero 6-flow if its underlying graph either contains a bridge, or is 3-edge-colorable, or is critical. In this paper, we improve and extend these results, and confirm Bouchet's conjecture for signed graphs with frustration number at most two, where the frustration number of a signed graph is the smallest number of vertices whose deletion leaves a balanced signed graph.     

Even and odd holes in cap‐free graphs

It is an old problem in graph theory to test whether a graph contains a chordless cycle of length greater than three (hole) with a specific parity (even, odd). Studying the structure of graphs without odd holes has obvious implications for Berge's strong perfect graph conjecture that states that a graph G is perfect if and only if neither G nor its complement contain an odd hole. Markossian, Gasparian, and Reed have proven that if neither G nor its complement contain an even hole, then G is β‐perfect. In this article, we extend the problem of testing whether G(V, E) contains a hole of a given parity to the case where each edge of G has a label odd or even. A subset of E is odd (resp. even) if it contains an odd (resp. even) number of odd edges. Graphs for which there exists a signing (i.e., a partition of E into odd and even edges) that makes every triangle odd and every hole even are called even‐signable. Graphs that can be signed so that every triangle is odd and every triangle is odd and every hole is odd are called odd‐signable. We derive from a theorem due to Truemper co‐NP characterizations of even‐signable and odd‐signable graphs. A graph is strongly even‐signable if it can be signed so that every cycle of length ≥ 4 with at most one chord is even and every triangle is odd. Clearly a strongly even‐signable graph is even‐signable as well. Graphs that can be signed so that cycles of length four with one chord are even and all other cycles with at most one chord are odd are called strongly odd‐signable. Every strongly odd‐signable graph is odd‐signable. We give co‐NP characterizations for both strongly even‐signable and strongly odd‐signable graphs. A cap is a hole together with a node, which is adjacent to exactly two adjacent nodes on the hole. We derive a decomposition theorem for graphs that contain no cap as induced subgraph (cap‐free graphs). Our theorem is analogous to the decomposition theorem of Burlet and Fonlupt for Meyniel graphs, a well‐studied subclass of cap‐free graphs. If a graph is strongly even‐signable or strongly odd‐signable, then it is cap‐free. In fact, strongly even‐signable graphs are those cap‐free graphs that are even‐signable. From our decomposition theorem, we derive decomposition results for strongly odd‐signable and strongly even‐signable graphs. These results lead to polynomial recognition algorithms for testing whether a graph belongs to one of these classes. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 289–308, 1999     

Signed graphs

A signed graph is a graph with a sign attached to each arc. This article introduces the matroids of signed graphs, which generalize both the polygon matroids and the even-circle (or unoriented cycle) matroids of ordinary graphs. The concepts of balance, switching, restriction and contraction, double covering graphs, and linear representation of signed graphs are treated in terms of the matroid, and a matrix-tree theorem for signed graphs is proved. The examples treated include the all-positive and all-negative graphs (whose matroids are the polygon and even-circle matroids), sign-symmetric graphs (related to the classical root systems), and signed complete graphs (equivalent to two-graphs).Replacing the sign group by an arbitrary group leads to voltage graphs. Most of our results on signed graphs extend to all voltage graphs.     

Graph theoretic approach to parallel gene assembly

We study parallel complexity of signed graphs motivated by the highly complex genetic recombination processes in ciliates. The molecular gene assembly operations have been modeled by operations of signed graphs, i.e., graphs where the vertices have a sign + or −. In the optimization problem for signed graphs one wishes to find the parallel complexity by which the graphs can be reduced to the empty graph. We relate parallel complexity to matchings in graphs for some natural graph classes, especially bipartite graphs. It is shown, for instance, that a bipartite graph G has parallel complexity one if and only if G has a unique perfect matching. We also formulate some open problems of this research topic.     

Nowhere zero flows in line graphs

Cai an Corneil (Discrete Math. 102 (1992) 103–106), proved that if a graph has a cycle double cover, then its line graph also has a cycle double cover, and that the validity of the cycle double cover conjecture on line graphs would imply the truth of the conjecture in general. In this note we investigate the conditions under which a graph G has a nowhere zero k-flow would imply that L(G), the line graph of G, also has a nowhere zero k-flow. The validity of Tutte's flow conjectures on line graphs would also imply the truth of these conjectures in general.