Minimum Integer Research Articles

Adjacent vertex distinguishing total coloring in split graphs

An adjacent vertex distinguishing (AVD-) total coloring of a graph G is a total coloring such that any two adjacent vertices u and v have distinct sets of colors, that is, C(u)≠C(v), where C(v) is the set of colors of the edges incident to v and the color of v. The adjacent vertex distinguishing (AVD)-total chromatic number of a graph G, χa″(G) is the minimum integer k such that there exists an AVD-total coloring of G using k colors. It is known that χa″(G)≥Δ+1, where Δ is the maximum degree of the graph. The AVD-total coloring conjecture states that for any graph G, χa″(G)≤Δ+3. In this paper, we study AVD-total coloring in split graphs. We verify the AVD-total coloring conjecture for split graphs and classify certain classes of split graphs according to their AVD-total chromatic number.

Twin-width II: small classes

The recently introduced twin-width of a graph \(G\) is the minimum integer \(d\) such that \(G\) has a \(d\)-contraction sequence, that is, a sequence of \(\left| V(G) \right|-1\) iterated vertex identifications for which the overall maximum number of red edges incident to a single vertex is at most \(d\), where a red edge appears between two sets of identified vertices if they are not homogeneous in \(G\) (not fully adjacent nor fully non-adjacent). We show that if a graph admits a \(d\)-contraction sequence, then it also has a linear-arity tree of \(f(d)\)-contractions, for some function \(f\). Informally if we accept to worsen the twin-width bound, we can choose the next contraction from a set of \(\Theta(\left| V(G) \right|)\) pairwise disjoint pairs of vertices. This has two main consequences. First it permits to show that every bounded twin-width class is small, i.e., has at most \(n!c^n\) graphs labeled by \([n]\), for some constant \(c\). This unifies and extends the same result for bounded treewidth graphs [Beineke and Pippert, JCT '69], proper subclasses of permutations graphs [Marcus and Tardos, JCTA '04], and proper minor-free classes [Norine et al., JCTB '06]. It implies in turn that bounded-degree graphs, interval graphs, and unit disk graphs have unbounded twin-width. The second consequence is an \(O(\log n)\)-adjacency labeling scheme for bounded twin-width graphs, confirming several cases of the implicit graph conjecture. We then explore the small conjecture that, conversely, every small hereditary class has bounded twin-width. The conjecture passes many tests. Inspired by sorting networks of logarithmic depth, we show that \(\log_{\Theta(\log \log d)}n\)-subdivisions of \(K_n\) (a small class when \(d\) is constant) have twin-width at most \(d\). We obtain a rather sharp converse with a surprisingly direct proof: the \(\log_{d+1}n\)-subdivision of \(K_n\) has twin-width at least \(d\). Secondly graphs with bounded stack or queue number (also small classes) have bounded twin-width. These sparse classes are surprisingly rich since they contain certain (small) classes of expanders. Thirdly we show that cubic expanders obtained by iterated random 2-lifts from \(K_4\) [Bilu and Linial, Combinatorica '06] also have bounded twin-width. These graphs are related to so-called separable permutations and also form a small class. We suggest a promising connection between the small conjecture and group theory. Finally we define a robust notion of sparse twin-width. We show that for a hereditary class \(\mathcal C\) of bounded twin-width the five following conditions are equivalent: every graph in \(\mathcal C\) (1) has no \(K_{t,t}\) subgraph for some fixed \(t\), (2) has an adjacency matrix without a \(d\)-by-\(d\) division with a 1 entry in each of the \(d^2\) cells for some fixed \(d\), (3) has at most linearly many edges, (4) the subgraph closure of \(\mathcal C\) has bounded twin-width, and (5) \(\mathcal C\) has bounded expansion. We discuss how sparse classes with similar behavior with respect to clique subdivisions compare to bounded sparse twin-width.Mathematics Subject Classifications: 68R10, 05C30, 05C48Keywords: Twin-width, small classes, expanders, clique subdivisions, sparsity

Total Rainbow Connection Number of Some Graph Operations

In a graph H with a total coloring, a path Q is a total rainbow if all elements in V(Q)∪E(Q), except for its end vertices, are assigned different colors. The total coloring of a graph H is a total rainbow connected coloring if, for any x,y∈V(H), there is a total rainbow path joining them. The total rainbow connection number trc(H) of H is the minimum integer r such that there is a total rainbow-connected coloring of H using r colors. In this paper, we study the total rainbow connection number of several graph operations (specifically, adding an edge, deleting an edge, and the Cartesian product) for which the total rainbow connection number is upper-bounded by a linear function of its radius.

Generalized signed graphs of large girth and large chromatic number

Assume G is a graph and S is a set of permutations of positive integers. An S-labelling of G is a pair (D,σ), where D is an orientation of G and σ:E(D)→S is a mapping which assigns to each arc e=(u,v) of D a permutation σe∈S. A proper k-colouring of (D,σ) is a mapping f:V(G)→[k]={1,2,...,k} such that σe(f(x))≠f(y) for each arc e=(x,y). We say S is non-trivial if for every positive integer i, there is a permutation π∈S such that π(i)=i, and S is transitive if for every pair (i,j) of positive integers, there is a permutation π∈S such that π(i)=j or π(j)=i. The S-chromatic number of a graph G is the minimum integer k such that any S-labelling (D,σ) of G has a proper k-colouring. This paper constructs, for any non-trivial set S of permutations of integers, for any integers k,g, a graph of girth at least g and S-chromatic number greater than k. We further prove that if S is a non-trivial set of permutations of positive integers, and 2≤k′≤k and g are positive integers, then the following two statements are equivalent: (1) there exists a graph of girth at least g, of chromatic number k′ and S-chromatic number greater than k, (2) for any k′-subset I of [k], there exist a,b∈I and π∈S such that a≠b and either π(a)=b or π(b)=a. As a consequence, there is a bipartite graph of arbitrary large girth and arbitrary large S-chromatic number if and only if S is transitive. In particular, there is a bipartite graph G of arbitrary large girth and arbitrary large group chromatic number.

Minimum Quantum Degrees for Isotropic Grassmannians in Types B and C

We give a formula in terms of Young diagrams to calculate the minimum positive integer d such that $$q^d$$ appears in the quantum product of two Schubert classes for the submaximal isotropic Grassmannians in types B and C. We do this by studying curve neighborhoods. We compute curve neighborhoods in several combinatorial models including k-strict partitions and a set of partitions where their inclusion is compatible with the Bruhat order.

Gallai–Ramsey number for K5 ${K}_{5}$

AbstractGiven a graph , the ‐colored Gallai–Ramsey number is defined to be the minimum integer such that every ‐coloring of the edges of the complete graph on vertices contains either a rainbow triangle or a monochromatic copy of Fox et al. conjectured the values of the Gallai–Ramsey numbers for complete graphs. Recently, this conjecture has been verified for the first open case, when . In this paper we attack the next case, when . Surprisingly it turns out, that the validity of the conjecture depends upon the (yet unknown) value of the Ramsey number . It is known that and conjectured that . If , then Fox et al.'s conjecture is true and we present a complete proof. If, however, , then Fox et al.'s conjecture is false, meaning that exactly one of these conjectures is true while the other is false. For the case when , we show lower and upper bounds for the Gallai–Ramsey number .

3-Vertices with fewest 2-neighbors in plane graphs with no long paths of 2-vertices

Let g(k,t) be the minimum integer such that every plane graph with girth g at least g(k,t), minimum degree δ=2 and no (k+1)-paths consisting of vertices of degree 2, where k≥1, has a 3-vertex with at least t neighbors of degree 2, where 1≤t≤3.In 2015, Jendrol' and Maceková proved g(1,1)≤7. Later on, Hudák et al. established g(1,3)=10, Jendrol', Maceková, Montassier, and Soták proved g(1,1)≥7, g(1,2)=8 and g(2,2)≥11, and we recently proved that g(2,2)=11 and g(2,3)=14.Thus g(k,t) is already known for k=1 and all t. In this paper, we prove that g(k,1)=3k+4, g(k,2)=3k+5, and g(k,3)=3k+8 whenever k≥2.

Sumsets and Veronese varieties

In this paper, to any subset mathcal {A}subset mathbb {Z}^{n} we explicitly associate a unique monomial projection Y_{n,d_{mathcal {A}}} of a Veronese variety, whose Hilbert function coincides with the cardinality of the t-fold sumsets tmathcal {A}. This link allows us to tackle the classical problem of determining the polynomial p_{mathcal {A}} in mathbb {Q}[t] such that |tmathcal {A}| = p_{mathcal {A}}(t) for all t ge t_0 and the minimum integer n_0(mathcal {A}) le t_0 for which this condition is satisfied, i.e. the so-called phase transition of |tmathcal {A}|. We use the Castelnuovo–Mumford regularity and the geometry of Y_{n,d_{mathcal {A}}} to describe the polynomial p_{mathcal {A}}(t) and to derive new bounds for n_0(mathcal {A}) under some technical assumptions on the convex hull of mathcal {A}; and vice versa we apply the theory of sumsets to obtain geometric information of the varieties Y_{n,d_{mathcal {A}}}.

Ramsey Numbers of Fans and Large Books

For graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum integer $N$ such that any red/blue edge coloring of $K_N$ contains either a red $G$ or a blue $H$. Let $G+H$ be the graph obtained from vertex disjoint $G$ and $H$ by adding new edges connecting $G$ and $H$ completely, $F_m=K_1+mK_2$ and $B_p(n)=K_p+nK_1$. It is shown $R(F_m,B_p(n))=2(n+p-1)+1$ for fixed $m, p$ and large $n$.

On 1-Subdivisions of Transitive Tournaments

The oriented Ramsey number $\vec{r}(H)$ for an acyclic digraph $H$ is the minimum integer $n$ such that any $n$-vertex tournament contains a copy of $H$ as a subgraph. We prove that the $1$-subdivision of the $k$-vertex transitive tournament $H_k$ satisfies $\vec{r}(H_k)= O(k^2\log\log k)$. This is tight up to multiplicative $\log\log k$-term.
 We also show that if $T$ is an $n$-vertex tournament with $\Delta^+(T)-\delta^+(T)= O(n/k) - k^2$, then $T$ contains a $1$-subdivision of $\vec{K}_k$, a complete $k$-vertex digraph with all possible $k(k-1)$ arcs. This is tight up to multiplicative constant.

Decomposing planar graphs into graphs with degree restrictions

AbstractGiven a graph , a decomposition of is a partition of its edges. A graph is ‐decomposable if its edge set can be partitioned into a ‐degenerate graph and a graph with maximum degree at most . For , we are interested in the minimum integer such that every planar graph is ‐decomposable. It was known that , , and . This paper proves that , and .

Ramsey number of multiple copies of stars

Let G and H be simple graphs. The Ramsey number r(G,H) is the minimum integer N such that any red-blue-coloring of edges of KN contains either a red copy of G or a blue copy of H. Let mK1,t denote m vertex-disjoint copies of K1,t. A lower bound is that r(mK1,t,nK1,s)≥m(t+1)+n−1. Burr, Erdős and Spencer proved that this bound is indeed the Ramsey number r(mK1,t,nK1,s) for t=s=3, m≥2 and m≥n. In this paper, we show that this bound is the Ramsey number r(mK1,t,nK1,s) for t≥s=3,m≥2 and m≥n. We also show that this bound is the Ramsey number r(mK1,t,nK1,s) for s≥4,t>s(s−1)2 and m>n.

Coloring squares of planar graphs with small maximum degree

For a graph $G$, by $\chi_2(G)$ we denote the minimum integer $k$, such that there is a $k$-coloring of the vertices of $G$ in which vertices at distance at most 2 receive distinct colors. Equivalently, $\chi_2(G)$ is the chromatic number of the square of $G$. In 1977 Wegner conjectured that if $G$ is planar and has maximum degree $\Delta$, then $\chi_2(G) \leq 7$ if $\Delta \leq 3$, $\chi_2(G) \leq \Delta+5$ if $4 \leq \Delta \leq 7$, and $\lfloor 3\Delta/2 \rfloor +1$ if $\Delta \geq 8$. Despite extensive work, the known upper bounds are quite far from the conjectured ones, especially for small values of $\Delta$. In this work we show that for every planar graph $G$ with maximum degree $\Delta$ it holds that $\chi_2(G) \leq 3\Delta+4$. This result provides the best known upper bound for $6 \leq \Delta \leq 14$.

Injective edge-coloring of subcubic graphs

An injective edge-coloring [Formula: see text] of a graph [Formula: see text] is an edge-coloring such that if [Formula: see text], [Formula: see text], and [Formula: see text] are three consecutive edges in [Formula: see text] (they are consecutive if they form a path or a cycle of length three), then [Formula: see text] and [Formula: see text] receive different colors. The minimum integer [Formula: see text] such that, [Formula: see text] has an injective edge-coloring with [Formula: see text] colors, is called the injective chromatic index of [Formula: see text] ([Formula: see text]). This parameter was introduced by Cardoso et al. [Injective coloring of graphs, Filomat 33(19) (2019) 6411–6423, arXiv:1510.02626] motivated by the Packet Radio Network problem. They proved that computing [Formula: see text] of a graph [Formula: see text] is NP-hard. We give new upper bounds for this parameter and we present the relationships of the injective edge-coloring with other colorings of graphs. We study the injective edge-coloring of some classes of subcubic graphs. We prove that a subcubic bipartite graph has an injective chromatic index bounded by [Formula: see text]. We also prove that if [Formula: see text] is a subcubic graph with maximum average degree less than [Formula: see text] (respectively, [Formula: see text]), then [Formula: see text] admits an injective edge-coloring with at most 4 (respectively, [Formula: see text]) colors. Moreover, we establish a tight upper bound for subcubic outerplanar graphs.

Weighted Scoring Committees

Weighted committees allow shareholders, party leaders, etc. to wield different numbers of votes or voting weights as they decide between multiple candidates by a given social choice method. We consider committees that apply scoring methods such as plurality, Borda, or antiplurality rule. Many different weights induce the same mapping from committee members’ preferences to winning candidates. The numbers of respective weight equivalence classes and hence of structurally distinct plurality committees, Borda commitees, etc. differ widely. There are 6, 51, and 5 plurality, Borda, and antiplurality committees, respectively, if three players choose between three candidates and up to 163 (229) committees for scoring rules in between plurality and Borda (Borda and antiplurality). A key implication is that plurality, Borda, and antiplurality rule are much less sensitive to weight changes than other scoring rules. We illustrate the geometry of weight equivalence classes, with a map of all Borda classes, and identify minimal integer representations.

On n-absorbing ideals of locally divided commutative rings

In 2011, David F. Anderson and Ayman Badawi defined n-absorbing ideals, and introduced the invariant ωR(I), which is the minimal integer n for which I is an n-absorbing ideal of a commutative ring R. We prove that the conjectures concerning n-absorbing ideals raised by Anderson and Badawi are true on locally divided rings.

Bounded VC-Dimension Implies the Schur-Erdős Conjecture

In 1916, Schur introduced the Ramsey number r(3; m), which is the minimum integer n > 1 such that for any m-coloring of the edges of the complete graph Kn, there is a monochromatic copy of K3. He showed that r(3; m) ≤ O(m!), and a simple construction demonstrates that r(3; m) ≥ 2Ω(m). An old conjecture of Erdős states that r(3; m) = 2Θ(m). In this note, we prove the conjecture for m-colorings with bounded VC-dimension, that is, for m-colorings with the property that the set system induced by the neighborhoods of the vertices with respect to each color class has bounded VC-dimension.

Almost all about light neighborhoods of 5-vertices in 3-polytopes with minimum degree 5

Let wΔ be the minimum integer W with the property that every 3-polytope with minimum degree 5 and maximum degree Δ has a vertex of degree 5 with the degree-sum (weight) of all vertices in its neighborhood at most W.Trivially, w5=30 and w6=35. In 1940, Lebesgue proved wΔ≤Δ+31 for all Δ≥5 and wΔ≤Δ+27 for Δ≥41.In 1998, the first Lebesgue's result was improved by Borodin and Woodall to wΔ≤Δ+30. This bound is sharp for Δ=7 due to Borodin (1992) and Jendrol' and Madaras (1996), Δ=9 due to Borodin and Ivanova (2013), Δ=10 due to Jendrol' and Madaras (1996), and Δ=12 due to Borodin and Woodall (1998).As for the second Lebesgue's bound, Borodin, Ivanova, and Jensen (2014) proved that wΔ=Δ+27 for Δ≥28, but w20≥48; the former fact was extended by Borodin and Ivanova (2016) to wΔ=Δ+27 for Δ≥24.In 2017, we proved wΔ≤Δ+29 whenever Δ≥13, and showed by constructions that w8=38, w11=41, and w13=42. Li, Rao, and Wang (2019) proved wΔ≤Δ+28 for Δ≥16.The purpose of this paper is to prove that wΔ=Δ+28 whenever 14≤Δ≤20. Thus wΔ remains unknown only for 21≤Δ≤23.

Signed colouring and list colouring of k‐chromatic graphs

AbstractA ‐colouring of a signed graph is a mapping such that for each edge , where is a symmetric integer set of size (i.e., implies that ). The signed chromatic number of a graph is the minimum integer such that for any signature of has a ‐colouring. Let be the maximum signed chromatic number of an ‐vertex ‐chromatic graph. This paper determines the value of for all positive integers . Then we study the list colouring of signed graphs. A list assignment of is called symmetric if is a symmetric integer set for each vertex . The weak signed choice number of a graph is defined to be the minimum integer such that for any symmetric ‐list assignment of , for any signature on , there is a proper ‐colouring of . We prove that the difference can be arbitrarily large. On the other hand, is bounded from above by twice the list vertex arboricity of . Using this result, we prove that , where is the complete ‐partite graph with each partite set of size 2.

Minimal group determinants for dicyclic groups

We determine the minimal non-trivial integer group determinant for the dicyclic group of order $4n$ when $n$ is odd. We also discuss the set of all integer group determinants for the dicyclic groups of order $4p$.