The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs.The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences.

ABSTRACTThe minimum positive co‐degree of a nonempty ‐graph , denoted , is the maximum such that if is an ‐set contained in a hyperedge of , then is contained in at least distinct hyperedges of . Given an ‐graph , we introduce the positive co‐degree Turán number as the maximum positive co‐degree over all ‐vertex ‐graphs that do not contain as a subhypergraph. In this paper, we concentrate on the behavior of for 3‐graphs . In particular, we determine asymptotics and bounds for several well‐known concrete 3‐graphs (e.g. and the Fano plane). We also show that, for ‐graphs, the limit exists, and “jumps” from 0 to , that is, it never takes on values in the interval . Moreover, we characterize which ‐graphs have . Our motivation comes primarily from the study of (ordinary) co‐degree Turán numbers where a number of results have been proved that inspire our results.

ABSTRACTAn oriented graph is converse invariant if, for any tournament , the number of copies of in is equal to that of its converse . El Sahili and Ghazo Hanna [J. Graph Theory 102 (2023), 684‐701] showed that any oriented graph with maximum degree at most 2 is converse invariant. They proposed a question: Can we characterize all converse invariant oriented graphs? In this paper, we introduce a digraph polynomial and employ it to give a necessary condition for an oriented graph to be converse invariant. This polynomial serves as a cornerstone in proving all the results presented in this paper. In particular, we characterize all orientations of trees with diameter at most 3 that are converse invariant. We also show that all orientations of regular graphs are not converse invariant if and have different degree sequences. In addition, in contrast to the findings of El Sahili and Ghazo Hanna, we prove that every connected graph with maximum degree at least 3, admits an orientation of such that is not converse invariant. We pose a new conjecture.

ABSTRACTGiven a hypergraph , the dual hypergraph of is the hypergraph of all minimal transversals of . The dual hypergraph is always Sperner, that is, no hyperedge contains another. A special case of Sperner hypergraphs are the conformal Sperner hypergraphs, which correspond to the families of maximal cliques of graphs. All these notions play an important role in many fields of mathematics and computer science, including combinatorics, algebra, database theory, and so on. Motivated by a question related to clique transversals in graphs, we study in this paper conformality of dual hypergraphs and prove several results related to the problem of recognizing this property. In particular, we show that the problem is in co‐NP and that it can be solved in polynomial time for hypergraphs of bounded dimension. For dimension 3, we show that the problem can be reduced to 2‐Satisfiability. Our approach has an application in algorithmic graph theory: we obtain a polynomial‐time algorithm for recognizing graphs in which all minimal transversals of maximal cliques have size at most , for any fixed .

ABSTRACTAn independent set of size in a finite undirected graph is a set of vertices of the graph, no two of which are connected by an edge. Let be the number of independent sets of size in the graph and let . In 1987, Alavi, Malde, Schwenk, and Erdős asked if the independent set sequence of a tree is unimodal (the sequence goes up and then down). This problem is still open. In 2006, Levit and Mandrescu showed that the last third of the independent set sequence of a tree is decreasing. We show that the first 46.8% of the independent set sequence of a random tree is increasing with (exponentially) high probability as the number of vertices goes to infinity. So, the question of Alavi, Malde, Schwenk, and Erdős is “four‐fifths true,” with high probability. We also show unimodality of the independent set sequence of Erdős–Rényi random graphs, when the expected degree of a single vertex is large (with [exponentially] high probability as the number of vertices in the graph goes to infinity, except for a small region near the mode). A weaker result is shown for random regular graphs. The structure of independent sets of size as varies is of interest in probability, statistical physics, combinatorics, and computer science.

ABSTRACTA ‐uniform hypergraph is ‐partite if can be partitioned into sets such that every edge in contains precisely one vertex from each . We call such a graph ‐balanced if for each . An independent set in is balanced if for each , and a coloring is balanced if each color class induces a balanced independent set in . In this paper, we provide a lower bound on the balanced independence number in terms of the average degree , and an upper bound on the balanced chromatic number in terms of the maximum degree . Our results recover those of recent work of Chakraborti for .

ABSTRACTThe square of a graph , denoted , has the same vertex set as and has an edge between two vertices if the distance between them in is at most 2. In general, for every graph . Charpentier (2014) asked whether if . But Hocquard, Kim, and Pierron (2019) answered his question negatively. For every even value of g, they constructed a 2‐degenerate graph such that . Note that if is a 2‐degenerate graph, then . Thus, we have that So, it was naturally asked whether there exists a constant such that if is a 2‐degenerate graph with . Recently, Cranston and Yu (2024) showed that if is a 2‐degenerate graph, and if is a 2‐degenerate graph with . We show that there exists a constant such that if is a 2‐degenerate graph with . This upper bound on is tight by the construction in Hocquard, Kim, and Pierron.

Abstract A graph is called odd (respectively, even) if every vertex has odd (respectively, even) degree. Gallai proved that every graph can be partitioned into two even induced subgraphs, or into an odd and an even induced subgraph. We refer to a partition into odd subgraphs as an odd colouring of . Scott proved that a connected graph admits an odd colouring if and only if it has an even number of vertices. We say that a graph is ‐odd colourable if it can be partitioned into at most odd induced subgraphs. The odd chromatic number of , denoted by , is the minimum integer for which is ‐odd colourable. We initiate the systematic study of odd colouring and odd chromatic number of graph classes. We first consider a question due to Scott, which states that every graph of even order has , for some positive constant , by proving that this is indeed the case if is restricted to having girth at least seven. We also show that any graph whose all components have even order satisfies , where is the maximum degree of . Next, we show that certain interesting classes have bounded odd chromatic number. Our main results in this direction are that interval graphs, graphs of bounded modular‐width all have bounded odd chromatic number. In particular, every even interval graph is 6‐odd colourable, and every even graph is ‐odd colourable, where is the modular width of a graph.