Let G be an n-vertex graph and let ℓ:V(G)→𝔽_2 prescribe degree parities. A set S⊆ V(G) is ℓ-admissible if every v∈ S has degree congruent to ℓ(v) modulo 2 in G[S]. Let h_ℓ(G) be the maximum order of an ℓ-admissible set, set f_oe(G):=min_ℓ h_ℓ(G), and write f_o(G):=h_1(G), where 1(v)=1 for every v∈ V(G). We prove three main results for graphs without isolated vertices. First, by extending Zeng's odd-cut method to arbitrary parity prescriptions an introducing a one-sided completion lemma, we show that h_ℓ(G)≥ n/6 for every ℓ. Consequently, f_oe(G)≥ n/6, improving the previous bound 2n/21. Second, for bipartite graphs we derive lower bounds on f_o(G) in terms of the 𝔽_2-rank of the bipartite adjacency matrix and combine them to obtain f_o(G)≥(1/4+1/256)n=65/256n. Thus, in the bipartite case, the factor 2 in Scott's bound f_o(G)≥ n/(2χ(G)) can be replaced by 128/65<2. Finally, writing α=α(G), a fourth-moment argument gives, for α≥2, f_o(G)≥2+log_3α/8 -1/4log_3log_3. We also construct bipartite graphs satisfying f_o(G)≤α(G)/2+log_2(α(G)+1)+1/2, showing that the logarithmic additive improvement over Scott's bound f_o(G)(G)/2 has the optimal order of magnitude.

A maximal independent set in a graph $G$ is an independent set that cannot be extended to a larger independent set by adding any vertex from $G$. This paper investigates the problem of determining the maximum number of maximal independent sets in terms of the matching number of a graph. We establish the maximum number of maximal independent sets for general graphs, connected graphs, triangle-free graphs, and connected triangle-free graphs with a given matching number, and characterize the extremal graphs achieving these maxima.

Determining an upper bound on s for vertex-primitive s-arc-transitive digraphs has been an open problem of considerable interest since a question asked by Praeger in 1990. Although much progress has been made and an upper bound is conjectured to be 2, a complete classification for s=2 remains out of reach. In this paper, we prove that the tight upper bound on s for finite vertex-primitive s-arc-transitive Cayley digraphs is exactly 2. Furthermore, we completely characterize the structure of these digraphs when s=2.

We study the first nonzero Steklov eigenvalue λ_2(T,δΩ) of the Dirichlet-to-Neumann operator on a finite tree T with leaf boundary δΩ, under a constraint on the diameter D. He and Hua [Calc. Var. PDE, 2022] showed that λ_2(T) ≤ 2/D for any tree of diameter D, with the even-diameter equality case fully characterized. For odd D, the geometric picture underlying the sharp configurations has remained unclear beyond diameter three. We determine this picture completely for all odd diameters D = 2r+1 ≥ 5. The sharp value of λ_2 is achieved on spider trees with nearly-equidistributed branch lengths, forming the family of generalized almost seesaw trees AS(r,q+2,c,t), prescribed by the arithmetic of n relative to ⌈ r/2 ⌉. Together with the results of He-Hua and Lin-Zhao [Bull. Lond. Math. Soc., 2025] for even diameters and diameter three, this completes the geometric classification for every diameter. The argument is based on a scalar root equation for one-center profiles, an inverse boundary quadratic form on boundary fluxes, and a reduction scheme from arbitrary trees to two-center profiles, and then to the one-center class. The inverse variational viewpoint may be regarded as a boundary analogue of the classical distance-matrix formalism for trees initiated by Graham and Lovász [Adv. Math., 1978].

For a digraph D, let (D) be the largest size of a vertex set no two of whose vertices lie in a common directed 2-cycle. Let f_2(a) be the least integer K such that every K-connected digraph D with (D)≤ a has a Hamilton cycle. In 1987, Jackson proved that f_2(a)≤ 2^a(a+2)! and asked for better bounds, noting that a linear bound might be possible. Kühn and Osthus later observed that even a polynomial bound would be interesting. In this short note, we prove the polynomial bound f_2(a)≤ 2a^3+2.

A trail is antidirected if its arcs alternate between forward and backward. A digraph D is antistrong if, for every ordered pair of distinct vertices x,y∈ V(D), it contains a forward antidirected (x,y)-trail. Bang-Jensen, Bessy, Jackson and Kriesell [J. Combin. Theory Ser. B 122 (2017), 68–90] introduced antistrong connectivity and posed two problems concerning mixed arc-disjoint spanning subdigraphs. In the first problem, one seeks an antistrong spanning subdigraph and an arc-disjoint strong spanning subdigraph. In the second, strong connectivity is replaced by the requirement that the underlying graph of the second subdigraph be 2-edge-connected. Bang-Jensen et al. asked whether each of the two problems can be solved in polynomial time. We prove that the two associated decision problems are NP-complete. The first remains NP-complete for digraphs with maximum out-degree at most four and maximum in-degree at most five. The second remains NP-complete even for oriented digraphs that are strong and antistrong, whose underlying graphs are 3-vertex-connected, and in which all but at most two vertices have both in-degree and out-degree at most four. In particular, the latter hardness result does not rely on digons.

ABSTRACT This paper establishes a relation between orientations and circular flows in signed graphs. It is proved that the circular flow index of a signed graph is strictly smaller than if and only if it admits a strongly connected ‐extended‐Tutte orientation. As applications, we utilize a unified approach to show that every ‐edge‐connected signed planar graph has its circular flow index strictly less than for . By duality, this provides upper bounds for circular chromatic numbers of signed planar graphs with given girth conditions, improving several known results. In particular, our results imply that every signed planar graph of girth at least 8 has its circular chromatic number strictly less than 3.

It is well known that the algebraic multiplicity of an eigenvalue of a graph (or real symmetric matrix) is equal to the dimension of its corresponding linear eigen-subspace, also known as the geometric multiplicity. However, for hypergraphs, the relationship between these two multiplicities remains an open problem. For a graph C = (V, E) and k >= 3, the kpower hypergraph C(k) is a k-uniform hypergraph obtained by adding k-2 new vertices to each edge of C, who always has non-real eigenvalues. In this paper, we determine the secondlargest modulus Lambda among the eigenvalues of C(k), which is indeed an eigenvalue of C(k). The projective eigenvariety V Lambda associated with Lambda is the set of the eigenvectors of C(k) corresponding to Lambda considered in the complex projective space. We show that the dimension of V Lambda is zero, i.e., there are finitely many eigenvectors corresponding to Lambda up to a scalar. We give both the algebraic multiplicity of Lambda and the total multiplicity of the eigenvector in V Lambda in terms of the number of the weakest edges of C. Our results show that these two multiplicities are equal. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let be a graph of genus with boundary . For , Lin and Zhao [J. Lond. Math. Soc. 112 (2025), Paper No. e70238] proved an upper bound for the first (non-trivial) Steklov eigenvalue of (G,delta Omega), and they posed the problem of determining a corresponding bound for graphs of genus g>0. In this paper, we prove an O(g|delta Omega|) bound for a bounded-degree graph of positive genus . Our result can be regarded as a discrete analogue of Kokarev's bound [Adv. Math. 258 (2014), 191-239], up to a constant factor.

A graph $G$ is called $C_k$-saturated if $G$ is $C_k$-free but $G+e$ not for any $e\in E(\overline{G})$. The saturation number of $C_k$, denoted $sat(n,C_k)$, is the minimum number of edges in a $C_k$-saturated graph on $n$ vertices. Finding the exact values of $sat(n,C_k)$ has been one of the most intriguing open problems in extremal graph theory. In this paper, we study the saturation number of $C_6$. We prove that ${4n}/{3}-2 \le sat(n,C_6) \le {(4n+1)}/{3}$ for $n\ge9$, which significantly improves the existing lower and upper bounds for $sat(n,C_6)$.

In this article, we present a unified approach for proving several Tur & aacute;n-type and generalized Tur & aacute;n-type problems, degree power problems, and extremal spectra problems on paths, cycles, and matchings. Specifically, we generalize classical results on cycles and matchings by Kopylov, Erd & odblac;s-Gallai, and Luo et al., respectively, and provide a positive resolution to an open problem originally proposed by Nikiforov. Moreover, we improve the spectral extremal results on paths, building on the work of Nikiforov, and Nikiforov and Yuan. Additionally, we provide a comprehensive solution to the connected version of the problem related to the degree power sum of a graph that contains no path on k vertices, a topic initially investigated by Caro and Yuster.

Constructing the maximum spanning tree T of an edge-weighted connected graph G is one of the important research topics in computer science and optimization, and the related research results have played an active role in practical applications. In this paper, we are concerned with the ratio of the weighted sum of a spanning tree T of G to the weighted sum of G, which we try to minimize. We propose an interesting theorem to simplify this problem and show that this optimal problem can be solved in polynomial time. Furthermore, we apply the optimal problem in chordal graphs.

A rainbow-free coloring of a k-uniform hypergraph H is a vertex-coloring which uses k colors but with the property that no edge of H attains all colors. Koerkamp and Z̆ivný showed that p=(k−1)(logn)/n is the threshold function for the existence of a rainbow-free coloring of the random k-uniform hypergraph Gk(n,p), and presented that the case when p is close to the threshold is open. In this paper, we give an answer to the question.

Let be an edge‐colored graph on vertices. The minimum color degree of , denoted by , is defined as the minimum number of colors assigned to the edges incident to a vertex in . In 2013, Li proved that an edge‐colored graph on vertices contains a rainbow triangle if . In this paper, we obtain several estimates on the number of rainbow triangles through one given vertex in . As a consequence, we prove counting results for rainbow triangles in edge‐colored graphs. One main theorem states that the number of rainbow triangles in is at least , which is best possible by considering the rainbow ‐partite Turán graph, where its order is divisible by . This means that there are rainbow triangles in if , and rainbow triangles in if when . Both results are tight in the sense of the order of the magnitude. We also prove a counting version of a previous theorem on rainbow triangles under a color neighborhood union condition due to Broersma et al., and an asymptotically tight color degree condition forcing a colored friendship subgraph (i.e., rainbow triangles sharing a common vertex).

For any positive integer k, the reconfiguration graph for all k-colorings of a graph G, denoted by R k （G）, is the graph where vertices represent the k-colorings of G, and two k-colorings are joined by an edge if they differ in color on exactly one vertex. Bonamy et al. established that for any 2-chromatic P 5 -free graph G,R k （G） is connected for each k≥3. On the other hand, Feghali and Merkel proved the existence of a 7p-chromatic P 5 -free graph G for every positive integer p, such that R 8p （G） is disconnected.In this paper, we offer a detailed classification of the connectivity of R k （G）conc(erning t-chromatic P 5 -free graphs G for cases t=3, and t≥4 with■. We demonstrate that R k （G） remains connected for each 3-chromatic P 5 -free graph G and each k≥4. Furthermore, for each t≥4 and ■.we provide a construction of a t-chromatic P 5 -free graph G with R k （G） being disconnected. This resolves a question posed by Feghali and Merkel.

A graph G is called F-saturated if G does not contain F as a subgraph (not necessarily induced) but the addition of any missing edge to G creates a copy of F. The saturation number of F, denoted by sat(n,F), is the minimum number of edges in an n-vertex F-saturated graph. Determining the saturation number of complete bipartite graphs is one of the most important problems in the study of saturation numbers. The value of sat(n,K2,2) was shown to be ⌊3n−52⌋ by Ollmann, and a shorter proof was later given by Tuza. For K2,3, there has been a series of study aiming to determine sat(n,K2,3) over the years. This was finally achieved by Chen who confirmed a conjecture of Bohman, Fonoberova, and Pikhurko that sat(n,K2,3)=2n−3 for all n≥5. Pikhurko and Schmitt conjectured that sat(n,K3,3)=(3+o(1))n. In this paper, for n≥9, we give an upper bound of 3n−9 for sat(n,K3,3), and prove that 3n−9 is also a lower bound when the minimum degree of a K3,3-saturated graph is 2 or 5, where it is trivial when the minimum degree is greater than 5.

Tutte's 3-flow conjecture states that every 4-edge-connected graph admits a nowhere-zero 3-flow. The planar case of Tutte's 3-flow conjecture is the classical Grötzsch's Theorem (1959). Steinberg and Younger (1989) further verified Tutte's 3-flow conjecture for projective planar graphs. In this paper we confirm Tutte's 3-flow conjecture for all toroidal graphs.

The planar Turán number of a graph H, denoted by ex__𝒫(n,H) , is the maximum number of edges in a planar graph on n vertices without containing H as a subgraph. This notion was introduced by Dowden in 2016 and has attracted quite some attention since then; those work mainly focus on finding ex__𝒫(n,H) when H is a cycle or Theta graph or H has maximum degree at least four. In this paper, we first completely determine the exact values of ex__𝒫(n,H) when H is a cubic graph. We then prove that ex__𝒫(n,2C_3)=⌈ 5n/2⌉ -5 for all n≥ 6 , and obtain the lower bounds of ex__𝒫(n,2C_k) for all n≥ 2k≥ 8 . Finally, we also completely determine the exact values of ex__𝒫(n,K_2,t) for all t≥ 3 and n≥ t+2 .

A strengthening of Jaeger's circular flow conjecture, restricted to planar graphs, asserts that every planar graph of odd girth at least 4k+1 admits a homomorphism to the odd cycle C2k+1, and the first case is verified and known as the famous Grötzsch theorem. In this paper, we prove analogous results for signed planar graphs: For k∈{2,3,4} every signed bipartite planar graph of negative girth at least 6k−4 admits a homomorphism to C−2k. Here the negative girth is the length of a shortest cycle with an odd number of negative edges. Note that the k=2 case was previously obtained in [J. Combin. Theory Ser. B, 153 (2022) 81–104] through a coloring method. Considering the duality between circular colorings and circular flows of planar graphs, our approach is based on the tools developed in the study of flows and group connectivity, and a potential method is applied in handling orientations with special boundaries for planar graphs. Furthermore, our results have several implications for the circular chromatic numbers of signed planar graphs with given girth conditions.

Given a planar graph family F ${\rm{ {\mathcal F} }}$, let exP(n,F) $e{x}_{{\mathscr{P}}}(n,{\mathscr{F}})$ and spexP(n,F) $spe{x}_{{\mathscr{P}}}(n,{\mathscr{F}})$ be the maximum size and maximum spectral radius over all n $n$-vertex F ${\rm{ {\mathcal F} }}$-free planar graphs, respectively. Let tCl $t{C}_{\ell }$ be the disjoint union of t $t$ copies of l $\ell $-cycles, and tC $t{\mathscr{C}}$ be the family of t $t$ vertex-disjoint cycles without length restriction. Tait and Tobin determined that K2+Pn-2 ${K}_{2}+{P}_{n-2}$ is the extremal spectral graph among all planar graphs with sufficiently large order n $n$, which implies the extremal graphs of both spexP(n,tCl) $spe{x}_{{\mathscr{P}}}(n,t{C}_{\ell })$ and spexP(n,tC) $spe{x}_{{\mathscr{P}}}(n,t{\mathscr{C}})$ for t >= 3 $t\ge 3$ are K2+Pn-2 ${K}_{2}+{P}_{n-2}$. In this paper, we first determine spexP(n,tCl) $spe{x}_{{\mathscr{P}}}(n,t{C}_{\ell })$ and spexP(n,tC) $spe{x}_{{\mathscr{P}}}(n,t{\mathscr{C}})$ and characterize the unique extremal graph for 1 <= t <= 2 $1\le t\le 2$, l >= 3 $\ell \ge 3$ and sufficiently large n $n$. Second, we obtain the exact values of exP(n,2C4) $e{x}_{{\mathscr{P}}}(n,2{C}_{4})$ and exP(n,2C) $e{x}_{{\mathscr{P}}}(n,2{\mathscr{C}})$, which solve a conjecture of Li for n >= 2661 $n\ge 2661$.