Ascending Subgraph Decompositions in Oriented Complete Balanced Tripartite Graphs

On the ascending subgraph decomposition problem for bipartite graphs

Let t and q be positive integers that satisfy C ( t + 1 , 2) ≤ q < C ( t + 2 , 2) and let G be a simple and finite graph of size q . G is said to have ascending subgraph decomposition (ASD) if G can be decomposed into t subgraphs H 1 ,H 2 , … ,H t without isolated vertices such that H i is isomorphic to a proper subgraph of H i +1 for 1 ≤ i ≤ t - 1 , where { E ( H 1 ) , … ,E ( H t ) } is a partition of E ( G ) . A graph that admits an ascending subgraph decomposition is called an ASD graph. In this paper, we introduce a new type of magic labeling based on the notion of ASD. Let G be an ASD graph and f : V ( G ) ∪ E ( G ) →{ 1 , 2 , … , | V ( G ) | + | E ( G ) |} be a bijection. The weight of a subgraph H i (1 ≤ i ≤ n ) is w ( H i ) = ∑ v ∈ V ( H i ) f ( v ) + ∑ e ∈ E ( H i ) f ( e ) . If the weight of each ascending subgraph is constant, say w ( H i ) = k , ∀ 1 ≤ i ≤ t , then f is called an ASD-magic labeling of G and G is called an ASD-magic graph. We present general properties of ASD-magic graphs and characterize certain classes of them.

Determining equitable total chromatic number for infinite classes of complete [formula omitted]-partite graphs

Balancedness of some subclasses of circular-arc graphs

Given a graph whose edges never fail but whose nodes fail independently of each other with a constant probability 1 − p, the reliability of a graph is defined to be the probability that the induced subgraph of the surviving nodes is connected. Let Ω (n, m) be the class of all graphs with n nodes and m edges. A graph G of Ω is said to be uniformly best in Ω, if for all choices of p, it is most reliable in the class of graphs, Ω. In this paper, the existing known set of uniformly best graphs is extended, and some closely related graphs are proved to be not the uniformly best graphs in their class. More precisely, we prove that for any positive integer b the complete tripartite graph K(b, b + 1, b + 2) is uniformly best in its class Ω(3b + 3, 3b2 + 6b + 2), while the complete tripartite graphs K(b, b + 1, b + i)(i > 2) are not the uniformly best in their classes Ω(3b + i + 1, 3b2 + 2(i + 1)b + i). © 2000 John Wiley & Sons, Inc.

The vertex arboricity va(G) of a graph G is the minimum number of subsets into which the vertex set V(G) can be partitioned so that each subset induces an acyclic subgraph. The fractional version of vertex arboricity is introduced in this paper. We determine fractional vertex arboricity for several classes of graphs, e.g., complete multipartite graphs, cycles, integer distance graphs, prisms and Peterson graph.

In this paper we consider a colouring version of the general position problem. The gp-chromatic number is the smallest number of colours needed to colour V(G) such that each colour class has the no-three-in-line property. We determine bounds on this colouring number in terms of the diameter, general position number, size, chromatic number, cochromatic number and total domination number and prove realisation results. We also determine it for several graph classes, including Kneser graphs K(n,2), line graphs of complete graphs, complete multipartite graphs, block graphs and Cartesian products. We also show that the gp-colouring problem is NP-complete.

Modularity has been introduced as a quality measure for graph partitioning. It has received considerable attention in several disciplines, especially complex systems. In order to better understand this measure from a graph theoretical point of view, we study the modularity of a variety of graph classes. We first consider simple graph classes such as tori and hypercubes. We show that these regular graph families have asymptotic modularity 1 (that is the maximum possible). We extend this result to the general class of unit ball graphs of bounded growth metrics. Our most striking result concerns trees with bounded degree which also appear to have asymptotic modularity 1. This last result can be extended to graphs with constant average degree and to some power-law graphs.

Graph classification is a critical task in analyzing graph data, with applications across various domains. While graph neural networks (GNNs) have achieved remarkable results, their ability to generalize across graphs of varying scales remains a challenge. Conventional models often perform well on large-scale graphs but struggle with distributions that are skewed towards small scales. Conversely, models tailored to address scale imbalances frequently prioritize small-scale graphs, leading to diminished performance in more balanced scenarios. To overcome these limitations, we introduce a Unbalanced-Balanced Representation Converter (U2B), which exhibits no explicit bias toward graph scales. U2B employs a two-step workflow: a distillation phase to extract base features from both node-level and graph-level representations, followed by a refinement phase to generate unbiased representations for improved balance. In the distillation phase, a static constraint guides node-level adjustments, improving the representation of nodes in small graphs. Simultaneously, a dynamic constraint in the graph-level process mitigates biases toward features from large graphs. To ensure harmony between the representations, a consistency alignment loss is introduced, aligning node-level and graph-level features to create more cohesive and balanced graph representations. Extensive experiments on multiple datasets show that U2B achieves competitive performance.

The Tur\'{a}n number for a graph $H$ is the least possible number of edges on an $n$-vertex graph with no copy of $H$ as a subgraph. For graphs $G$ and $H$, the \emph{anti-Ramsey number}, denoted $\ar(G,H)$, is the minimum number of colors $d$ such that for any edge coloring with $d$ colors there exists a rainbow copy of $H$. The concept of anti-Ramsey theory was first introduced by Erd\H{o}s, Simonovits, and S\'{o}s in 1973 as a dual problem to traditional Ramsey theory. This thesis investigates Tur\'{a}n theory and anti-Ramsey theory for specific graph families, including cliques in multipartite graphs and matchings in hypergraphs. In Chapter 2, the anti-Ramsey number for disjoint cliques in complete, balanced multipartite graphs is given for certain sizes of cliques and multipartite graphs. In particular, the value $\ar(K_{r\times n}, nK_r)$ is given for all integers $n,r\ge 3$ and $\ar(K_{n,n,n}, 3K_r)$ is given for all integers $n,r\ge 3$. Chapter 3 generalizes Tur\'{a}n and anti-Ramsey theory to the context of hypergraphs. The Tur\'{a}n number for $3$-matchings in complete $3$-uniform hypergraphs is determined. Next, two proofs of the value for the anti-Ramsey number for $4$-matchings in complete hypergraphs are presented. One relies on the Tur\'{a}n number for $3$-matchings, and the other utilizes a method called progressive induction. Chapter 4 features additional work related to combinatorics more generally.

Multithreshold multipartite graphs with small parts

The paper provides a complete solution to the existence problem of two orthogonal one-factorizations of a complete balanced multipartite graph K_{ptimes q}. In particular, new classes of Howell designs are constructed.

Let the star on n vertices, namely K 1, n − 1 be denoted by S n . If every two coloring of the edges of a complete balanced multipartite graph K j × s there is a copy of S n in the first color or a copy of S m in the second color, then we will say K j × s → ( S n , S m ) . The size Ramsey multipartite number m j ( S n , S m ) is the smallest natural number s such that K j × s → ( S n , S m ) . In this paper, we obtain the exact values of the size Ramsey numbers m j ( S n , S m ) for n , m ≥ 3 and j ≥ 3 .

Ascending subgraph decompositions of regular graphs

Several implicit methods to infer horizontal gene transfer (HGT) focus on pairs of genes that have diverged only after the divergence of the two species in which the genes reside. This situation defines the edge set of a graph, the later-divergence-time (LDT) graph, whose vertices correspond to genes colored by their species. We investigate these graphs in the setting of relaxed scenarios, i.e., evolutionary scenarios that encompass all commonly used variants of duplication-transfer-loss scenarios in the literature. We characterize LDT graphs as a subclass of properly vertex-colored cographs, and provide a polynomial-time recognition algorithm as well as an algorithm to construct a relaxed scenario that explains a given LDT. An edge in an LDT graph implies that the two corresponding genes are separated by at least one HGT event. The converse is not true, however. We show that the complete xenology relation is described by an rs-Fitch graph, i.e., a complete multipartite graph satisfying constraints on the vertex coloring. This class of vertex-colored graphs is also recognizable in polynomial time. We finally address the question “how much information about all HGT events is contained in LDT graphs” with the help of simulations of evolutionary scenarios with a wide range of duplication, loss, and HGT events. In particular, we show that a simple greedy graph editing scheme can be used to efficiently detect HGT events that are implicitly contained in LDT graphs.