Journal of Graph Theory

For a finite set of integers such that the first few gaps between its consecutive elements equal a a, while the remaining gaps equal b b, we study dense packings of its translates on the line. We obtain an explicit lower bound on the corresponding optimal density, conjecture its tightness, and prove it in case one of the gap lengths, a a or b b, appears only once. This is equivalent to a Motzkin problem on the independence ratio of certain integer distance graphs.

The minimum positive co‐degree of a nonempty r r‐graph H H, denoted δr−1+(H) ${\delta }_{r-1}^{+}(H)$, is the maximum k k such that if S S is an (r−1) $(r-1)$‐set contained in a hyperedge of H H, then S S is contained in at least k k distinct hyperedges of H H. Given an r r‐graph F F, we introduce the positive co‐degree Turán number co+ex(n,F) ${\text{co}}^{+}\text{ex}(n,F)$ as the maximum positive co‐degree δr−1+(H) ${\delta }_{r-1}^{+}(H)$ over all n n‐vertex r r‐graphs H H that do not contain F F as a subhypergraph. In this paper, we concentrate on the behavior of co+ex(n,F) ${\text{co}}^{+}\text{ex}(n,F)$ for 3‐graphs F F. In particular, we determine asymptotics and bounds for several well‐known concrete 3‐graphs F F (e.g. K4− ${K}_{4}^{-}$ and the Fano plane). We also show that, for r r‐graphs, the limit γ+(F)≔limn→∞co+ex(n,F)n ${\gamma }^{+}(F):= {\mathrm{lim}}_{n\to \infty }\frac{{\text{co}}^{+}\text{ex}{(}n{,}F{)}}{n}$ exists, and “jumps” from 0 to 1/r 1/r, that is, it never takes on values in the interval (0,1/r) (0,1/r). Moreover, we characterize which r r‐graphs F F have γ+(F)=0 ${\gamma }^{+}(F)=0$. 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.

We show that every commutative idempotent monoid (a.k.a. lattice) is the endomorphism monoid of a subcubic graph. This solves a problem of Babai and Pultr and the degree bound is best‐possible. On the other hand, we show that no class excluding a minor can have all commutative idempotent monoids among its endomorphism monoids. As a by‐product, we prove that monoids can be represented by graphs of bounded expansion (reproving a result of Nešetřil and Ossona de Mendez) and k k‐cancellative monoids can be represented by graphs of bounded degree. Finally, we show that not all completely regular monoids can be represented by graphs excluding topological minor (strengthening a result of Babai and Pultr).

We prove that for any graph G G, the total chromatic number of G G is at most Δ ( G ) + 2 | V ( G ) | Δ ( G ) + 1 {\rm{\Delta }}(G)+2\unicode{x02308}\frac{|V(G)|}{{\rm{\Delta }}(G)+1}\unicode{x02309}. This saves one color in comparison with the result of Hind from 1992. In particular, our result says that if Δ ( G ) ≥ 1 2 | V ( G ) | ${\rm{\Delta }}(G)\ge \frac{1}{2}|V(G)|$, then G G has a total coloring using at most Δ ( G ) + 4 ${\rm{\Delta }}(G)+4$ colors. When G G is regular and has a sufficient number of vertices, we can actually save an additional two colors. Specifically, we prove that for any 0 < ε < 1 $0\lt \varepsilon \lt 1$, there exists n 0 ∈ N ${n}_{0}\in {\mathbb{N}}$ such that: if G G is an r r‐regular graph on n ≥ n 0 $n\ge {n}_{0}$ vertices with r ≥ 1 2 ( 1 + ε ) n $r\ge \frac{1}{2}(1+\varepsilon )n$, then χ T ( G ) ≤ Δ ( G ) + 2 ${\chi }_{{\rm{T}}}(G)\le {\rm{\Delta }}(G)+2$. This confirms the Total Coloring Conjecture for such graphs G G.

Barnette's conjecture asserts that every cubic 3‐connected plane bipartite graph is hamiltonian. Although, in general, the problem is still open, some partial results are known. In particular, let us call a face of a plane graph big (small) if it has at least six edges (it has four edges, respectively). Goodey proved for a 3‐connected bipartite cubic plane graph P P, that if all big faces in P P have exactly six edges, then P P is hamiltonian. In this paper, we prove that the same is true under the condition that no face in P P has more than four big neighbours. We also prove, that if each vertex in P P is incident both with a small and a big face, then P P has at least 2 k ${2}^{k}$ different Hamilton cycles, where k = ∣ B ∣ − 2 4 Δ ( B ) − 7 k=\unicode{x02308}\frac{| B| -2}{4{\rm{\Delta }}(B)-7}\unicode{x02309}, ∣ B ∣ $| B|$ is the number of big faces in P P and Δ ( B ) ${\rm{\Delta }}(B)$ is the maximum size of faces in P P.

Let G G be a graph in which each edge is assigned one of the colours 1,2,…,m $1,2,\ldots ,m$, and let Γ ${\rm{\Gamma }}$ be a subgroup of Sm ${S}_{m}$. The operation of switching at a vertex x x of G G with respect to an element π $\pi$ of Γ ${\rm{\Gamma }}$ permutes the colours of the edges incident with x x according to π $\pi$. We investigate the complexity of whether there exists a sequence of switches that transforms a given m m‐edge‐coloured graph G G so that it has an edge‐colour‐preserving homomorphism to a fixed m m‐edge‐coloured graph H H and give a dichotomy theorem in the case that Γ ${\rm{\Gamma }}$ acts transitively.

A cycle is said to be directed if all its arcs have the same direction. Otherwise, it is said to be nondirected. A strong tournament is a tournament containing a directed path from any vertex to any other vertex. A tournament that is not strong is said to be reducible. Rosenfeld conjectured that there exists an integer n0≥9 ${n}_{0}\ge 9$ such that every tournament of order n≥n0 $n\ge {n}_{0}$ contains any Hamiltonian nondirected cycle. Havet proved this conjecture for n≥68 $n\ge 68$ and for reducible tournaments for n≥9 $n\ge 9$. Finding non‐Hamiltonian cycles seems more simple. Thomason proved that any tournament of order n≥16 $n\ge 16$ contains any nondirected cycle of order n−2 $n-2$. This implies the existence of cycles of order m m, 14≤m≤n−2 $14\le m\le n-2$ in every tournament of order n≥16 $n\ge 16$. He said that the result is probably true for n≥5 $n\ge 5$. In this paper, we prove the existence of any nondirected cycle of order m m, 3≤m≤n−1 $3\le m\le n-1$ in every tournament of order n≥4 $n\ge 4$ unless five exceptions.

For integers k>ℓ≥0 $k\gt \ell \ge 0$, a graph G G is (k,ℓ) $(k,\ell )$‐stable if α(G−S)≥α(G)−ℓ $\alpha (G-S)\ge \alpha (G)-\ell$ for every S⊆V(G) $S\subseteq V(G)$ with ∣S∣=k $| S| =k$. A recent result of Dong and Wu states that every (k,ℓ) $(k,\ell )$‐stable graph G G satisfies α(G)≤⌊(∣V(G)∣−k+1)/2⌋+ℓ $\alpha (G)\le \lfloor (| V(G)| -k+1)/2\rfloor +\ell$. A (k,ℓ) $(k,\ell )$‐stable graph G G is tight if α(G)=⌊(∣V(G)∣−k+1)/2⌋+ℓ $\alpha (G)=\lfloor (| V(G)| -k+1)/2\rfloor +\ell$; and q q‐tight for some integer q≥0 $q\ge 0$ if α(G)=⌊(∣V(G)∣−k+1)/2⌋+ℓ−q $\alpha (G)=\lfloor (| V(G)| -k+1)/2\rfloor +\ell -q$. In this paper, we first prove that for all k≥24 $k\ge 24$, the only tight (k,0) (k,0)‐stable graphs are Kk+1 ${K}_{k+1}$ and Kk+2 ${K}_{k+2}$, answering a question of Dong and Luo. We then prove that for all nonnegative integers k,ℓ,q $k,\ell ,q$ with k≥3ℓ+3 $k\ge 3\ell +3$, every q q‐tight (k,ℓ) $(k,\ell )$‐stable graph has at most k−3ℓ−3+23(ℓ+2q+4)2 $k-3\ell -3+{2}^{3{(\ell +2q+4)}^{2}}$ vertices, answering a question of Dong and Luo in the negative.

A bipartite graph G = ( V , E ) G=(V,E) with V = V 1 ∪ V 2 $V={V}_{1}\cup {V}_{2}$ is biregular if all the vertices of each stable set, V 1 ${V}_{1}$ and V 2 ${V}_{2}$, have the same degree, r r and s s, respectively. This paper studies difference sets derived from both Abelian and non‐Abelian groups. From them, we propose some constructions of bipartite biregular graphs with diameter d = 3 d=3 and asymptotically optimal order for given degrees r r and s s, meaning that asymptotically the order approaches a fixed multiple of the Moore bound. Moreover, we find some biMoore graphs, that is, bipartite biregular graphs that attain the Moore bound.

Modulo flow is a powerful tool in the study of flows in both ordinary graphs and signed graphs. For ordinary graphs, Tutte showed that a graph admits a nowhere‐zero k k‐flow if and only if it admits a nowhere‐zero Zk ${{\mathbb{Z}}}_{k}$‐flow. However, such equivalence does not hold any more for signed graphs. Mačajova and Škoviera [SIAM Journal of Discrete Mathematics 31 (2017) 1937–1952] proved that every flow‐admissible signed graph with a nowhere‐zero Z2 ${{\mathbb{Z}}}_{2}$‐flow admits a nowhere‐zero 4‐flow. DeVos et al. [Journal of Combinatorial Theory Series B 149 (2021) 198–221] proved that every flow‐admissible signed graph admits a nowhere‐zero 11‐flow by converting certain special nowhere‐zero Z6 ${{\mathbb{Z}}}_{6}$‐flows into integer flows, and as a key step, they showed that every signed graph with a nowhere‐zero Z3 ${{\mathbb{Z}}}_{3}$‐flow admits a nowhere‐zero 5‐flow. In this paper, we study how to convert Z4 ${{\mathbb{Z}}}_{4}$‐flows and Z5 ${{\mathbb{Z}}}_{5}$‐flows into integer flows by proving the following two results: (1) every flow‐admissible signed graph with a nowhere‐zero Z4 ${{\mathbb{Z}}}_{4}$‐flow admits a nowhere‐zero 8‐flow; (2) every bridgeless signed graph with a nowhere‐zero Z5 ${{\mathbb{Z}}}_{5}$‐flow admits a nowhere‐zero 7‐flow. Combining known results, it follows that every flow‐admissible signed graph with a nowhere‐zero Zk ${{\mathbb{Z}}}_{k}$‐flow admits a nowhere‐zero 2k 2k‐flow for each integer k≥2 $k\ge 2$.

In 2023, Bang‐Jensen, Havet and Yeo [J. Graph Theory 102 (2023) 578‐606] conjectured that every ‐strong semicomplete digraph contains a hamiltonian cycle avoiding any prescribed set of arcs, which was inspired by the result of Fraisse and Thomassen that, every ‐strong tournament contains a hamiltonian cycle avoiding any prescribed set of arcs. In this paper, we prove the conjecture.

A subgraph of a graph with maximum degree is ‐ overfull if . Clearly, if contains a ‐overfull subgraph, then its chromatic index is . However, the converse is not true, as demonstrated by the Petersen graph. Nevertheless, three families of graphs are conjectured to satisfy the converse statement: (1) graphs with (the Overfull Conjecture of Chetwynd and Hilton), (2) planar graphs (Seymour's Exact Conjecture), and (3) graphs whose subgraph induced on the set of maximum degree vertices is the union of vertex‐disjoint cycles (the Core Conjecture of Hilton and Zhao). Over the past decades, these conjectures have been central to the study of edge coloring in simple graphs. Progress had been slow until recently, when the Core Conjecture was confirmed by the authors in 2024. This breakthrough was achieved by extending Vizing's classical fan technique to two larger families of trees: the pseudo‐multifan and the lollipop. This paper investigates the properties of these two structures, forming part of the theoretical foundation used to prove the Core Conjecture. We anticipate that these developments will provide insights into verifying the Overfull Conjecture for graphs where the subgraph induced by maximum‐degree vertices has relatively small maximum degree.

Let be a graph on vertices, with complement . The spectral gap of the transition probability matrix of a random walk on is used to estimate how fast the random walk becomes stationary. We prove that the larger spectral gap of and is . Moreover, if all degrees are and , then the larger spectral gap of and is . We also show that if the maximum degree is or if is a join of two graphs, then the spectral gap of is . Finally, we provide a family of connected graphs with connected complements such that the larger spectral gap of and is .

We exhibit non‐switching‐isomorphic signed graphs that share a common underlying graph and common chromatic polynomials, thereby answering a question posed by Zaslavsky. We introduce a new pair of bivariate chromatic polynomials that generalises the chromatic polynomials of signed graphs. We establish recursive dominating‐vertex deletion formulae for these bivariate chromatic polynomials. As an application, we demonstrate that for a certain family of signed threshold graphs, isomorphism can be characterised by the equality of bivariate chromatic polynomials.

In this paper, we give two extremal results on vertex disjoint‐directed cycles in tournaments and bipartite tournaments. Let and be two integers. The first result is that for every strong tournament , with a minimum out‐degree of at least with , any vertex disjoint‐directed cycle, which has a length of at least in , has the same length if and only if and is isomorphic to . The second result is that for each strong bipartite tournament , with a minimum out‐degree of at least with being even, any vertex disjoint‐directed cycle, each of which has a length of at least in , has the same length if and only if is isomorphic to a member of . Our results generalize some results of Tan and of Chen and Chang, and in a sense, extend several results of Bang‐Jensen et al. of Ma et al. as well as of Wang et al.

We say that a graph is ‐edge‐Hamilton‐connected if has a Hamilton cycle containing all edges of for any with such that is a linear forest. In 2012, Kužel et al. conjectured that every 4‐connected line graph is 2‐edge‐Hamilton‐connected, and proved that it is equivalent to Thomassen's conjecture stating that every 4‐connected line graph is Hamiltonian. In this paper, we prove that for every ‐connected line graph is ‐edge‐Hamilton‐connected.

An inversion of a tournament is obtained by reversing the direction of all edges with both endpoints in some set of vertices. Let be the minimum length of a sequence of inversions using sets of size at most that result in the transitive tournament. Let be the maximum of taken over ‐vertex tournaments. It is well known that and it was recently proved by Alon et al. that . In these two extreme cases ( and ), random tournaments are extremal objects. It is proved that is not attained by random tournaments when and conjectured that is (only) attained by (quasi)random tournaments. It is further proved that and , where for all and for all .

We study extensions of Turán Theorem in edge‐weighted settings. A particular case of interest is when constraints on the weight of an edge come from the order of the largest clique containing it. These problems are motivated by Ramsey‐Turán type problems. Some of our proofs are based on the method of graph Lagrangians, while the other proofs use flag algebras. Using these results, we prove several new upper bounds on the Ramsey‐Turán density of cliques. Other applications of our results are in a recent paper of Balogh, Chen, McCourt, and Murley.

Tutte proved that 4‐connected planar graphs are Hamiltonian. It is unknown if there is an analogous result on 1‐planar graphs. In this paper, we characterize 4‐connected 1‐planar chordal graphs and show that all such graphs are Hamiltonian‐connected. A crucial tool used in our proof is a characteristic of 1‐planar 4‐trees.

In this article, we extend the duality relation between face colorings and integer flows of graphs on orientable surfaces in Tutte's flow theory to both orientable and nonorientable surfaces and study Bouchet's 6‐flow conjecture from point of embeddings of graphs on surfaces. Consequently, we verify Bouchet's conjecture for a family of embedded graphs, which have a crosscap‐contractible circuit.

A separating system of a graph is a family of subgraphs of for which the following holds: for all distinct edges and of , there exists an element in that contains but not . Recently, it has been shown that every graph of order admits a separating system consisting of paths, improving the previous almost linear bound of , and settling conjectures posed by Balogh, Csaba, Martin, and Pluhár and by Falgas‐Ravry, Kittipassorn, Korándi, Letzter, and Narayanan. We investigate a natural generalization of these results to subdivisions of cliques, showing that every graph admits both a separating system consisting of edges and cycles and a separating system consisting of edges and subdivisions of .

The recognition of Pfaffian bipartite graphs in polynomial time has been obtained, but this fact is still unknown for Pfaffian nonbipartite graphs. For a path and a cycle , the Pfaffian graphs of Cartesian products and were characterized by Lu and Zhang in 2014. Recently, Li and Wang characterized the Pfaffian graph for and the Pfaffian graph for is bipartite. However, the question of characterizing the Pfaffian graph is still open. In this paper, we try to investigate this question for the graph with a perfect matching. We first prove that () and are Pfaffian if and only if is an odd path or an even cycle, respectively. After showing that is not Pfaffian if G is nonbipartite, we obtain the characterization of the Pfaffian graph in terms of the forbidden subgraphs of .

Given a finite Lie incidence geometry, which is either a polar space of rank at least 3 or a strong parapolar space of symplectic rank at least 4 and diameter at most 4, or the parapolar space arising from the line Grassmannian of a projective space of dimension at least 4, we show that its point graph is determined by its local structure. This follows from a more general result, which classifies graphs whose local structure can vary over all local structures of the point graphs of the aforementioned geometries. In particular, this characterises the strongly regular graphs arising from the line Grassmannian of a finite projective space, from the half spin geometry related to the quadric Q + ( 10 , q ) and from the exceptional group of type E 6 ( q ) by their local structure.

A widely open conjecture proposed by Bollobás, Erdős, and Tuza in the early 1990s states that for any n‐vertex graph G, if the independence number α ( G ) = Ω ( n ), then there is a subset T ⊆ V ( G ) with ∣ T ∣ = o ( n ) such that T intersects all maximum independent sets of G. In this study, we prove that this conjecture holds for graphs that do not contain an induced K s , t for fixed t ≥ s. Our proof leverages the probabilistic method at an appropriate juncture.

Let G be a graph of order n. For an integer k ≥ 2, a partition P of V ( G ) is called a k‐proper partition of G if every P ∈ P induces a k‐connected subgraph of G. This concept was introduced by Ferrara et al., and Borozan et al. gave minimum degree conditions for the existence of a k‐proper partition. In particular, when k = 2, they proved that if δ ( G ) ≥ n, then G has a 2‐proper partition P with ∣ P ∣ ≤ n − 1 δ ( G ). Later, Chen et al. extended the result by giving a minimum degree sum condition for the existence of a 2‐proper partition. In this paper, we introduce two new invariants of graphs σ * ( G ) and α * ( G ), which are defined from degree sum of particular independent sets. Our result is that if σ * ( G ) ≥ n, then with some exceptions, G has a 2‐proper partition P with ∣ P ∣ ≤ α * ( G ). We completely determine exceptional graphs. This result implies both of results by Borozan et al. and by Chen et al. Moreover, we obtain a minimum degree product condition for the existence of a 2‐proper partition as a corollary of our result.