Papers

Let $G_n$ be the graph on $\{2,\ldots,n\}$ in which two integers are adjacent when they have a common divisor greater than one. We determine the complete asymptotic expansion of its Havel–Hakimi residue $\R(G_n)$, confirming a leading-constant prediction of Staton recorded in Fajtlowicz's Written on the Wall. If $A=\sum_{k=2}^{\infty}(\log k)/(k^2(k-1))$, then the first two terms are $\R(G_n)=(\zeta(2)-1)n/\log n+(\zeta(2)-1-A)n/\log^2n +O(n/\log^3n)$. More precisely, the difference between $\R(G_n)$ and the prime-vertex contribution to the Caro–Wei sum is $O_\beta(n\exp\{-(\log n)^\beta\})$ for every fixed $0<\beta<1/2$; this estimate yields every coefficient in the expansion. The upper bound follows from a degree-preserving realization in which almost all relevant prime vertices are partitioned into cliques. We also prove that the unlabeled graph determines $\pi(n)$ through its simplicial true-twin classes. Stable inverses for weighted sums of the resulting degree-class counts give criteria equivalent to the Riemann hypothesis, including one involving only the Caro–Wei sum. An exact local-defect identity additionally reduces the conjectured sharp $+2$ residue bound to explicit prefix estimates.

For a fixed integer $\ell \ge 2$, we study what values of chromatic index and chromatic number can be attained by some $\ell$-fold cyclic cover of a loopless multigraph. For edge-coloring, we first investigate the density, a fundamental lower bound for the chromatic index, and show that the density of every $\ell$-fold cyclic cover of a graph $G$ is at most that of $G$. We further prove that if $\ell$ is even, then the spectrum of chromatic indices over all $\ell$-fold cyclic covers of $G$ contains every integer between $\Delta(G)$ and $\chi'(G)$. When $\ell$ is odd, the chromatic-index spectrum need not be complete in general; for edge-chromatic critical graphs, we determine exactly which values are attainable. For vertex-coloring, we prove that if $\chi(G)\ge 3$, then the spectrum of chromatic numbers over all $\ell$-fold cyclic covers of $G$ contains every integer between $3$ and $\chi(G)$. Moreover, this spectrum contains $2$ if and only if $G$ is bipartite or $\ell$ is even.

We show that, up to isometry, there is exactly one compact hyperbolic Coxeter 6-polytope with 10 facets, the polytope $P_{6,10}$ attributed to Bugaenko. Together with results of Felikson-Tumarkin ($d \ge 7$) and of Burcroff and Ma-Zheng ($d = 4, 5$), this completes the classification of compact hyperbolic Coxeter $d$-polytopes with $d+4$ facets. The proof is computer assisted. Affine Gale duality applied to the complete database of order types on 10 points yields 387 combinatorial types, of which Lannér's classification excludes 83. For the remaining 304, an exhaustive search over Coxeter labellings, with no a priori bound on the dihedral angles, leaves a single realizable Gram matrix. Every rejection is certified in exact arithmetic, and completeness of the search is certified independently by DRAT proofs checked by drat-trim. The same code reproduces the known censuses in dimensions 4 and 5. Code, data and certificates are publicly available.

A dominating set $S$ of a graph $G$ is a locating-dominating set (LDS) if, for each pair of distinct vertices not in~$S$, their neighbourhoods in $S$ are distinct. Finding a minimum-cardinality LDS in finite graphs is a well-known NP-hard problem. On infinite graphs, this problem naturally generalises to finding an LDS of minimum density. While density bounds have been widely studied for specific infinite regular grids, no computational complexity results exist for infinite graphs. We prove that the minimum-density LDS problem in infinite $\mathbb{Z}$-periodic graphs with a finite period is NP-hard. This result bridges the gap between cardinality minimization on finite graphs and density minimization on infinite graphs via a rigorous periodic reduction. Furthermore, our approach can be adapted to establish NP-hardness for related structural problems on infinite periodic graphs.

Let $\mathfrak{B}_n$ denote the hyperoctahedral group. The type $B$ coinvariant rings $R_{\mathfrak{B}_n}^{(k,j)}$ are quotients of the ring of polynomials in $k$ sets of $n$ commuting variables and $j$ sets of $n$ anticommuting variables by the ideal generated by the diagonal $\mathfrak{B}_n$-invariants without constant term. Building upon the work of Kim–Rhoades (2022), we give an explicit formula for the bigraded Frobenius series of $R_{\mathfrak{B}_n}^{(0,2)}$: the bigraded multiplicity of each irreducible $\mathfrak{B}_n$-character is a single Schur polynomial, so $R_{\mathfrak{B}_n}^{(0,2)}$ is multiplicity-free as a $\operatorname{GL}_2 \times \mathfrak{B}_n$-module. We then determine that the trigraded multiplicity of the sign character of $R_{\mathfrak{B}_n}^{(0,3)}$ is given by a single Schur function. Finally, for all $k$ and $j$, we determine the multiplicity of the standard character in the type $A$ coinvariant ring $R_{n}^{(k,j)}$, as well as the multiplicities of the characters indexed by the bipartitions $((n-1),(1))$ and $((n-1,1),\varnothing)$ in $R_{\mathfrak{B}_n}^{(k,j)}$. These are the first nontrivial characters established for all $(k,j)$ in either of types $A$ or $B$.

We study sets of (measurable) returns in countable groups $G$, namely sets of the form $\{g\in G:\mu(A\cap T_gA)>0\}$ arising from measure-preserving actions. Extending a result of Bergelson, we show that sets of returns in $G\times G$ contain subsets of the form $B\times B$, where $B$ is large with respect to suitable notions of largeness that remain meaningful even for non-amenable groups. As a consequence, if $G$ is amenable, then every sufficiently large subset $A\subseteq G\times G$ satisfies $B\times B\subseteq AA^{-1}$ for some large set $B\subseteq G$. We also investigate when sets of returns in $G$ contain product sets $BB$ with $B$ large. In contrast with the Cartesian-product phenomenon above, this problem is considerably subtler in non-abelian groups and is closely connected to `symmetric correlation functions', namely functions of the form $g\mapsto \mu(T_g^{-1}A\cap T_gA)$. We use this connection to show that, for broad classes of amenable groups - including finitely generated nilpotent groups and certain solvable non-nilpotent groups, every sufficiently large set $A\subseteq G$ contains a large subset $B$ satisfying $BB\subseteq AA^{-1}$. Finally, we establish polynomial analogues of these results for finitely generated nilpotent groups, extending earlier work of Bergelson and Ruzsa.

This paper provides formulas for Koornwinder polynomials in analogy with the creation formula, the alcove walk formula and the non-attacking fillings formula for the type $GL_n$ Macdonald polynomials. We state the creation formula in terms of the divided-difference operators used in Schubert calculus, and we use a box-greedy reduced word to reformulate the alcove walk formula in terms of uncompressed set-valued tableaux. Then two types of compression, “around-the-end compression” and “across-the-$0$-gap compression”, are used to derive a formula for Koornwinder polynomials in terms of compressed set-valued tableaux. Throughout we work in the full generality of relative Koornwinder polynomials, which are the analogues of the permuted basement Macdonald polynomials used in the type $GL_n$ case.

We develop a local-to-global differential framework for additive decomposition problems involving multiplicative subgroups of prime fields. Starting from Hanson–Petridis-type auxiliary polynomials, we use degree bounds, in the spirit of Stepanov's method, to lift local coefficient relations at their roots to global differential identities. This yields a unified treatment of \[ H=A+B,\qquad H\cup\{0\}=A-A,\qquad H=A\mathbin{\widehat{+}} A,\qquad H\cup\{0\}=A\mathbin{\widehat{+}} A, \] where $H$ is a proper multiplicative subgroup. This circle of problems is motivated by Sárközy's conjecture on the additive irreducibility of nonzero quadratic residues and its generalizations to multiplicative subgroups. Rudnev and Tyrrell recently classified all decompositions $H=A+B$, building on the approach introduced by Hanson–Petridis and further developed by Kalmynin. Our framework gives a new polynomial proof of the Rudnev–Tyrrell classification and substantially streamlines the existing proofs: it gives an independent proof of Kalmynin's equal-size theorem and reduces the classification to direct coefficient comparisons, avoiding the residue-theoretic input and more elaborate arithmetic analysis of earlier proofs. It also yields a streamlined proof of Kalmynin's resolution of a conjecture of Lev and Sonn on $H\cup\{0\}=A-A$. For the two restricted-sumset problems, we obtain complete classifications, substantially improving earlier results of Shkredov and Yip. We also establish some stability refinements.

For a non-decreasing sequence $S=(a_1,a_2,\ldots,a_r)$ of positive integers, an $S$-packing coloring of a graph $G$ is a partition of $V(G)$ into sets $A_1,\ldots,A_r$ such that any two distinct vertices in $A_i$ are at distance greater than $a_i$, for every $i\in\{1,\ldots,r\}$. Gastineau and Togni [Discrete Math. 339 (2016), 2461–2470] asked whether every subcubic graph, except the Petersen graph, is $(1,1,2,3)$-packing colorable. In this paper, we prove that every claw-free subcubic graph is $(1,1,2,3)$-packing colorable. Moreover, we show that every connected claw-free subcubic graph, except a single graph $\mathcal{H}$, is $(1,1,3,3,3)$-packing colorable, thereby confirming a conjecture of the first two authors. Both results are best possible. Our proofs rely on a structural framework based on the skeleton and core graphs of a claw-free subcubic graph, together with a Hall-type matching argument that reduces the construction of suitable $3$-packings to a matching problem in an auxiliary bipartite graph.

In a recent breakthrough, OpenAI proved that the $k$-color Ramsey number of the triangle $C_3$ grows super-exponentially, more precisely, they proved that $R_k(C_3)\ge k^{k/3-o(k)}$. In this short note, we present a modification of their recursive construction that works for multicolor Ramsey numbers of fixed odd cycles. More precisely, for $p\ge 1$, let $\mathcal{O}_p=\{C_3,C_5,\ldots,C_{2p+1}\}$. We show that \[ R_k(\mathcal{O}_p)\ge (\log^{(p-1)}k)^{k/3-o(k)} \] for every fixed $p$, where $\log^{(p-1)}$ denotes the $(p-1)$-fold iterated logarithm. This immediately implies that for every fixed odd cycle, the multicolor Ramsey number is superexponential in the number of colors. The presented proof was found autonomously by ChatGPT 5.6 Pro/Sol.

Conlon and Lee asked for strongly dominating graphs beyond norming graphs and even paths. We construct a two-parameter family of pairwise non-isomorphic $2$-connected strongly dominating graphs that are not seminorming, and hence lie outside the two classes of examples previously identified for signed strong domination. The construction uses cyclic amalgamation of two-rooted blocks. For root-reversible blocks, we characterize the generation of all even cyclic amalgams by local even-Schatten inequalities for transfer operators. We determine this criterion for $K_{2,m}$, with the roots in the part of size $m$: it holds exactly when $m$ is even. We also classify the connected outerplanar strongly dominating graphs and the connected root-reversible outerplanar blocks satisfying the universal cyclic criterion.

A famous conjecture of Erdős and Hajnal (1969) states that for every integer $g\ge 4$ there is a smallest function $f_g:\mathbb{N}\to\mathbb{N}$ such that every graph of chromatic number at least $f_g(k)$ contains a subgraph of chromatic number $k$ and girth at least $g$. So far, this has only been proved for $g=4$ by Rödl (1977), with $f_4(k)$ bounded by a tower of height $\Theta(k^2\log k)$. We exhibit a surprising connection between finding high-chromatic subgraphs of large odd-girth (avoiding short odd cycles) and lower-bounding multicolor Ramsey numbers of odd cycles. Using this connection, we prove that for every odd $g\ge 5$ there is a function $h_g:\mathbb{N}\to\mathbb{N}$ growing as a power tower of height $\frac{g-3}{2}$ such that every graph of chromatic number at least $h_g(k)$ contains a subgraph of chromatic number at least $k$ and odd-girth at least $g$. This proves a conjecture of Mohar and Wu (2018), addresses a question of Erdős and Hajnal (1975), and for $g=5$ improves Rödl's bound on $f_4(k)$ to a single-exponential. We extend this to a much more general meta-theorem which applies to many graph parameters: if $f$ is the fractional chromatic number, the Hall ratio, or the strict vector chromatic number (Lovász-Theta-function of the complement), then for every $k,g\in\mathbb{N}$, every graph $G$ with sufficiently large $f(G)$ contains a subgraph $G'$ of odd-girth at least $g$ with $f(G')\ge k$. The key Ramsey-theoretic ingredient is a new lower bound on Ramsey numbers of odd cycles. For $p\ge 1$, let $\mathcal{O}_p=\{C_3,C_5,\ldots,C_{2p+1}\}$. We show that $R_k(\mathcal{O}_p)\ge(\log^{(p-1)}k)^{k/3-o(k)}$ for every fixed $p$, where $\log^{(p-1)}$ denotes the $(p-1)$-fold iterated logarithm. This yields the first superexponential lower bound on multicolor Ramsey numbers of fixed odd cycles, and extends the recent breakthrough by OpenAI for triangles.

A simple min-max theorem is formulated and proved for the smallest modification (measured in $l_1$-norm) of an input cost function $w_0$ that makes a target arborescence $F_0$ of a digraph a cheapest arborescence. The constructive proof gives rise to a polynomial time algorithm for computing both a minimizer cost function on the primal side and a maximizer dual object in the min-max formula.

For an elliptic curve $E$ defined over a field $K$ of characteristic $0$ with $\operatorname{End}_K \! E \cong \mathbb{Z}$, we classify which isogeny graphs $\mathcal{G}(E/K)$ can occur. We first show that $\mathcal{G}(E/K)$ decomposes as a weak Cartesian product of its $p$-primary isogeny graphs, one for each prime $p$, thereby reducing the problem to classifying $p$-primary isogeny graphs. We then show that each such graph is isomorphic, as an edge-weighted graph, to a member of an explicit family of edge-weighted graphs $\mathcal{H}_{p^k}^r$ and $\mathcal{H}_{p^{\infty,+}}^r$, every member of which occurs as a $p$-primary isogeny graph except for $\mathcal{H}_{2^k}^0$ for $k\ge 2$. The proof relies on a detailed study of the $p$-adic Galois representation attached to $E$, through which we identify each graph with a subgroup of $\operatorname*{GL}\nolimits_{2}(\mathbb{Z}_{p})$. More generally, we identify subgroups of $\operatorname*{GL}\nolimits_{2}(\widehat{\mathbb{Z}})$ for each possible isogeny graph and describe their corresponding modular curves, completing, in the genus $0$ case, the explicit parameterization of isogeny graphs via parameterized isogenous families of elliptic curves. We also introduce the $p$-blooming invariant $\mathfrak{I}_p(E/K)$, an isogeny class invariant determining the value of $r$ in the $p$-primary isogeny graph, and show that elliptic curves over fields with a real embedding attain the smallest possible value. As applications, we characterize the isogeny graphs of elliptic curves with potential complex multiplication; give an algorithm for determining the isogeny graph from the adelic Galois representation; recover the classification of rational isogeny graphs; and, under GRH, classify the isogeny graphs occurring over certain number fields.

In this paper, we study the Minkowski decomposability of symmetric edge polytopes $P_G^\pm$ of a finite simple graph $G$ on vertex set $[n]$. More precisely, we give a complete characterization of graphs whose symmetric edge polytopes are Minkowski decomposable. We prove that $P_G^\pm$ is Minkowski decomposable if and only if $G$ is one of the three complete multipartite graphs: $K_n$, $K_{2,n-2}$, or $K_{1,1,n-2}$. In other words, if $G$ does not belong to these three families, then $P_G^\pm$ is Minkowski indecomposable.

For an integer $k\ge2$, let $\lambda_k(G)$ denote the $k$th largest adjacency eigenvalue of a graph $G$. For every graph $G$ on $n$ vertices and every $2 \leq k \leq n$, we prove \[ \lambda_k(G) \le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1. \] Our bound is tight for $k\in\{2,3,4,8,24\}$. We obtain it by reducing the graph-eigenvalue problem to an extremal problem for orthogonal projections and then applying the general upper bound on the absolute projection constant $\gamma(r)$ due to Deręgowska and Lewandowska. We also give an alternative proof of their bound by repairing the Gegenbauer-polynomial argument of König and Tomczak-Jaegermann. The resulting slack identity yields a strict improvement in every even dimension $r\ge4$ for which $r+2$ is not a perfect square.

A famous result in the theory of combinatorial polynomials is the real-rootedness of the type $D$ Eulerian polynomial $D_n(x)$, which was originally conjectured by Brenti in 1994. By constructing a set of compatible polynomials over $s$-inversion sequences, Savage and Visontai proved this conjecture in 2013. Using matrices preserving interlacing properties of nonnegative polynomial sequences, Bränden also established the real-rootedness of $D_n(x)$. Combining Hermite-Biehler theorem and a result of Borcea and Brändén on Hurwitz stability, Yang and Zhang gave another proof of the real-rootedness of $D_n(x)$. By constructing half Eulerian polynomials of type $D$, Hyatt reproved Brenti's conjecture. As originally suggested by Brenti in 1994, it is possible that the real-rootedness of $D_n(x)$ may be established by using a more precise knowledge of the location of zeros of the types $A$ and $B$ Eulerian polynomials. In this paper, we add more details to the first proof of the real-rootedness of $D_n(x)$ that was provided by the author in 2012, which yields the half interlacing property among the types $A,B$ and $D$ Eulerian polynomials.

The existence of Hamilton cycles in connected vertex-transitive graphs is a core open problem in algebraic graph theory, originating from Lovász's 1969 conjecture. All connected vertex-transitive graphs of order $pq$ are known to be Hamiltonian except the Petersen graph, and primitive graph of order $2pq$ are resolved except the Coxeter graph. This paper considers connected vertex-transitive graphs of order $2pq$ where every transitive automorphism subgroup admits a maximal intransitive normal subgroup inducing prime-length orbits. We prove that all such graphs contain a Hamilton cycle, with no new exceptions beyond the already characterized non-qualifying graphs. This result covers a large non-quasiprimitive graphs of order $2pq$, advancing the full resolution of the $2pq$.

Let $G$ be an $n$-vertex graph and let $\ell:V(G)\to\mathbb{F}_2$ prescribe degree parities. A set $S\subseteq V(G)$ is $\ell$-admissible if every $v\in S$ has degree congruent to $\ell(v)$ modulo $2$ in $G[S]$. Let $h_\ell(G)$ be the maximum order of an $\ell$-admissible set, set $f_{\mathrm{oe}}(G):=\min_\ell h_\ell(G)$, and write $f_o(G):=h_{\mathbf{1}}(G)$, where $\mathbf{1}(v)=1$ for every $v\in V(G).$ 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 $h_\ell(G)\ge n/6$ for every $\ell$. Consequently, $f_{\mathrm{oe}}(G)\ge n/6$, improving the previous bound $2n/21$. Second, for bipartite graphs we derive lower bounds on $f_o(G)$ in terms of the $\mathbb{F}_2$-rank of the bipartite adjacency matrix and combine them to obtain \[ f_o(G)\ge \left(\frac14+\frac1{256}\right)n=\frac{65}{256}n. \] Thus, in the bipartite case, the factor $2$ in Scott's bound $f_o(G)\ge n/(2\chi(G))$ can be replaced by $128/65<2$. Finally, writing $\alpha=\alpha(G)$, a fourth-moment argument gives, for $\alpha\ge2$, \[ f_o(G)\ge \frac{\alpha}{2}+\frac{\log_3\alpha}{8} -\frac14\log_3\log_3\sqrt{\alpha}. \] We also construct bipartite graphs satisfying \[ f_o(G)\le \frac{\alpha(G)}2+\log_2\!\bigl(\alpha(G)+1\bigr)+\frac12, \] showing that the logarithmic additive improvement over Scott's bound $f_o(G)\ge\alpha(G)/2$ has the optimal order of magnitude.

DP-coloring (or correspondence coloring) is a generalization of list coloring that has been widely studied since its introduction by Dvořák and Postle in 2015. As the analogue of $P(G,q)$, the chromatic polynomial of a graph $G$, the DP color function of $G$, denoted by $P_{DP}(G,q)$, counts the minimum number of DP-colorings over all $q$-fold covers of $G$. It follows that $P_{DP}(G,q) \leq P(G,q)$. It is known that there are graphs for which $P_{DP}(G,q) < P(G,q)$ for all sufficiently large $q$; in fact, all bipartite graphs containing a cycle have this property. A fundamental open question about DP color functions asks whether, for every graph $G$, there exist $N \in \mathbb{N}$ and a polynomial $p$ such that $P_{DP}(G,q) = p(q)$ whenever $q \geq N$. In this paper we answer this question affirmatively for all bipartite graphs. Specifically, if $G$ is an $n$-vertex bipartite graph with $c$ components, then $P_{DP}(G,q) = (-1)^{n-c}q^c \;T_G(1-q,1)$ for all sufficiently large $q$, where $T_G(x,y)$ is the Tutte polynomial of $G$. The ideas we develop also yield an asymptotic formula for $P(G,q)-P_{DP}(G,q)$ whenever the girth of $G$ is even.

Let $T_n=K_{1,1,n}$, and let $P_n(x)$ denote the Kazhdan–Lusztig polynomial of its graphic matroid. We prove that, whenever $n\ge2$ and $0\le\lambda\le n/2$, the polynomial $P_n(x)+\lambda x$ has exactly $\lfloor n/2\rfloor$ zeros, all of which are negative and simple. In particular, the Kazhdan–Lusztig polynomials of the graphic matroids of $T_n$ and $K_{2,n}$ are real-rooted. We also prove that, for $n\ge2$, the common polynomial $Z_{T_n}(x)=Z_{K_{2,n}}(x)$ has $n+1$ distinct negative zeros. The proofs use a common rational transformation, reducing the Kazhdan–Lusztig case to alternating sign evaluations at the zeros of a Chebyshev polynomial and the $Z$-polynomial case to a unit-circle criterion for self-inversive polynomials.

Let \(\cG_1,\ldots,\cG_k\) be fixed vertex-ordered graphs, each containing at least one edge. The ordered Ramsey number \(\oR(\cG_1,\ldots,\cG_k)\) is the least integer \(N\) such that every \(k\)-edge-coloring of the ordered complete graph \(\cK_N\) contains an order-preserving copy of \(\cG_i\) in color \(i\) for some \(i\in[k]\). For positive weights \(\blambda=(\lambda_1,\ldots,\lambda_k)\), let \(\oM_{\blambda}(n;\cG_1,\ldots,\cG_k)\) denote the minimum weighted number of correctly colored, order-preserving copies of the target graphs over all \(k\)-edge-colorings of \(\cK_n\). When \(\blambda=\bf{1}\), \(\oM_{\bf{1}}(n;\cG_1,\ldots,\cG_k)=\oM(n;\cG_1,\ldots,\cG_k)\) is called the ordered Ramsey multiplicity. In this paper, we first establish the amplification inequality \[ \oM_{\blambda}(n;\cG_1,\ldots,\cG_k) \ge \oM_{\blambda}(t;\cG_1,\ldots,\cG_k) \frac{\binom{n}{\hmin}}{\binom{t}{\hmin}}, \] where $h_i=v(\cG_i),\hmin=\min_{i\in[k]}h_i$, and $n\ge t\ge\oR(\cG_1,\ldots,\cG_k)$. Let $\cS_{r,s}$ be the ordered star whose center has $r-1$ leaves to its left and $s-1$ leaves to its right, and let $\bB_m$ be the family of all ordered perfect matchings on $[2m]$ containing the edge $\{1,2m\}$. We apply the amplification inequality to obtain the multiplicity lower bounds for ordered stars and ordered perfect matchings. We then obtain the upper bound $\oM_{\boldsymbol\lambda} (n;\cS_{r_1,s_1},\cS_{r_2,s_2}) \le \min\{\lambda_1 B_{h_1}(n),\lambda_2 B_{h_2}(n)\}$ by constructions, where $B_{h_i}(n):= \binom{\lfloor n/2\rfloor}{h_i} + \binom{\lceil n/2\rceil}{h_i}$ and $h_i=r_i+s_i-1$ for $i\in [2]$. We also derive a random-coloring upper bound for ordered stars and prove \[\oM(n; \bB_m,\bB_m) \le \binom{n}{2m} \frac{(2m-2)!}{2^{2m-2}(m-1)!}.\] Finally, we establish a regularity-based lifting theorem for ordered colorings.

For a group $\Omega$, the associated power graph $P(\Omega)$ is defined as the graph whose vertices are the elements of $\Omega$, with two distinct vertices $u,v\in \Omega$ being adjacent if either $u=v^m$ or $v=u^n$ for some $m,n \in \mathbb{N}$. In this paper, we completely characterise the structure of the power graph associated with the class of metacyclic groups. Building on this structural description, we derive explicit expressions for the characteristic polynomials of the adjacency, Laplacian, and signless Laplacian matrices. Moreover, we obtain lower and upper bounds for the spectral radii of the adjacency and signless Laplacian matrices.

We prove that for all fixed $k\geq 4$, any $N$ vertex graph with no independent set of size $n$ and $N\geq \Omega(n^{k-1}/\log^{k-2}n)$ contains at least $$ \Omega\bigg(\binom Nk \Big(\frac{\log n}{n}\Big)^{\binom k2}/\log n\bigg) $$ cliques of order $k$, and for $k\geq 5$ this is best possible conditional on the known upper bounds for $r(k,n)$. This is also true and tight for $k=2$ by Turán's Theorem and for $k=3$ by a result of Bohman and Mubayi. We show the bound is also tight for $k=4$. We obtain other supersaturation results using the same methods.

We investigate $t$-intersecting families in direct-product set systems obtained by prescribing the number of selected elements in each part of a partitioned ground set. For a finite union of layers, we prove an Erdős–Ko–Rado result under coordinatewise linear part-size conditions. As an application, we establish a new range of parameters for which a conjecture of Frankl et al.\ [J. Combin. Theory Ser. A 155 (2018), 493–502] holds. Under an explicit polynomial large-part hypothesis, we also characterize the maximum nontrivial $t$-intersecting families for the single-layer setting. In particular, for $t=1$, this answers the problem of Kwan et al.\ [J. Combin. Theory Ser. A 156 (2018), 44–60] asking for a classification of all extremal families, including the possible non-shifted maximizers.

Let $k\geq2$ be an integer. A $1\bmod k$ edge-coloring of a graph $G$ is an edge-coloring in which every nonzero degree in each color class is congruent to $1$ modulo $k$. Let $\chi'_k(G)$ denote the minimum number of colors required, and let $\chi'_k$ be the supremum of $\chi'_k(G)$ over all finite simple graphs $G$. Botler, Colucci, and Kohayakawa conjectured that there exists an absolute constant $C$ such that $\chi'_k(G)\leq k+C$ for every $k$ and every $G$. We disprove this conjecture, even within the class of bipartite graphs. More precisely, for all integers $c\geq0$ and $k\geq3c+2$, we construct a finite simple bipartite graph $G_{k,c}$ satisfying $\chi'_k(G_{k,c})=k+c+1$. Consequently, $\chi'_k\geq k+\lfloor(k+1)/3\rfloor$ for every $k\geq2$. For $k_m=2\cdot3^{m-1}$, we give an affine-hyperplane construction of a finite simple bipartite graph $G_m$ satisfying $\Delta(G_m)=\chi'_{k_m}(G_m)=3^m=3k_m/2$. More generally, for every sufficiently large $k$, we construct a finite simple bipartite graph $G_k$ such that $\Delta(G_k)=\chi'_k(G_k)\geq3k/2-10(k\log k)^{1/3}$. Our proofs combine a codegree obstruction with explicit cyclic and affine-geometric constructions and a structured random perturbation.

These lecture notes are based on three courses given at Nagoya University. Their purpose is to give a beginner-friendly introduction to multiple zeta values and several of their variants, such as finite and symmetric multiple zeta values, q-analogues of multiple zeta values, and multiple Eisenstein series. These notes will be updated in the future.

We give a common counterexample to two product-structure conjectures in extremal graph theory. More precisely, we construct a fixed nonempty finite family $\mathcal L$ with $p(\mathcal L)=2$ such that, for some $c>0$, \[ \operatorname{ex}(n,\mathcal L)>t_2(n)+cn^{3/2} \] for every sufficiently large $n$. Nevertheless, at every such order there is an $\mathcal L$-extremal graph whose complement is connected and which therefore admits no decomposition as a join of two nonempty graphs. This superlinear surplus also forces the decomposition family of $\mathcal L$ to contain no forest. The construction uses endpoint-injective repair with a finite obstruction family admitting an exact extremal formula and equality classification.

We prove a graph-theoretical characterisation of finitely generated subgroups of Thompson's group $V$: a finitely generated group embeds in $V$ if and only if it admits a faithful context-free action, or equivalently if it belongs to the class CF-TR of transition groups of context-free graphs recently introduced by Matucci and the three last authors. Using this characterisation, we prove results in different directions: - All known examples of groups with co-context-free Word Problem do embed in $V$, providing evidence towards Lehnert's conjecture. - Each finitely generated subgroup of $V$ is either virtually abelian, or contains a free non-abelian semigroup. It follows that groups of intermediate growth do not embed in Thompson's $V$. We further study the relation between transition groups defined by graphs that are limits or covers of each others, and prove properties of transition groups of context-free graphs of polynomial growth. Finally, we prove that the Basilica and Hanoï Towers groups do not embed in $V$. This uses the geometry of Schreier graphs of the natural actions of these groups and of Thompson's $V$.

Sharma and Panda recently proved that every bipartite graph with a perfect matching has property (P); that is, it admits a non-singular real symmetric matrix with support graph G for which every vertex is a P -vertex. In this paper, we extend their result from bipartite graphs to arbitrary graphs. To this end, we introduce the notion of a P - vertex covering and define the P -vertex covering number p(G) as the minimum number of non-singular matrices in S(G) needed so that every vertex of G is a P -vertex of at least one of them. Given a maximal matching of G, we partition the vertex set into the vertices saturated by the matching and the remaining vertices, which necessarily form an independent set. We then construct separate matrices covering these two classes of vertices. We use the Implicit Function Theorem as a perturbation tool to establish the desired result.