Complete R-partite Graph Research Articles

Counting deranged matchings

Let pm(G) denote the number of perfect matchings of a graph G, and let Kr×2n/r denote the complete r-partite graph where each part has size 2n/r. Johnson, Kayll, and Palmer conjectured that for any perfect matching M of Kr×2n/r, we have for 2n divisible by rpm(Kr×2n/r−M)pm(Kr×2n/r)∼e−r/(2r−2).This conjecture can be viewed as a common generalization of counting the number of derangements on n letters, and of counting the number of deranged matchings of K2n. We prove this conjecture. In fact, we prove the stronger result that if R is a uniformly random perfect matching of Kr×2n/r, then the number of edges that R has in common with M converges to a Poisson distribution with parameter r2r−2.

The change of Seidel energy of 5-partite Turán graph due to edge deletion

Let G be a simple undirected graph. The Seidel energy of G, denoted by Es(G), is the sum of absolute values of all Seidel eigenvalues of G. A complete r-partite graph with n vertices is called r-partite Turán graph, denoted by T(n,r), if the number of vertices of each part is either ⌊nr⌋ or ⌈nr⌉. In 2021, Tian et al. proposed a problem that for any edge e of Turán graph T(n,r) with r≥4, does there exist a sufficiently large integer n0 such that Es(T(n,r))<Es(T(n,r)−e) whenever n≥n0. In this paper, we prove that the Seidel energy of T(n,5) of order n≥5 is always increased when an edge is deleted.

The sum of the k largest distance eigenvalues of graphs

Motivated by progresses on study of eigenvalue sum in adjacency matrix and Laplacian matrix, in this paper, we focus on the sum of the k largest distance eigenvalues of graphs. Denote by Sk(D(G))=λ1(D(G))+⋯+λk(D(G)) the sum of the k largest distance eigenvalues of G. We initially determine the extremal graph attaining the minimum Sk(D(G)) among all threshold graphs except for complete graphs. Besides, we prove that the complete r-partite graph attains the minimum Sk(D(G)) among all graphs with chromatic number χ(G)=r and we also give the lower bounds on Sk(D(G)) when G is Kr+1-free, which are extensions of previous results from Lin (2019) and Lu & Lu (2022), respectively.

On a conjecture of spectral extremal problems

For a simple graph F, let Ex(n,F) and Exsp(n,F) denote the set of graphs with the maximum number of edges and the set of graphs with the maximum spectral radius in an n-vertex graph without any copy of the graph F, respectively. The Turán graph Tn,r is the complete r-partite graph on n vertices where its part sizes are as equal as possible. Cioabă, Desai and Tait [The spectral radius of graphs with no odd wheels, European J. Combin., 99 (2022) 103420] posed the following conjecture: Let F be any graph such that the graphs in Ex(n,F) are Turán graphs plus O(1) edges. Then Exsp(n,F)⊂Ex(n,F) for sufficiently large n. In this paper we consider the graph F such that the graphs in Ex(n,F) are obtained from Tn,r by adding O(1) edges, and prove that if G has the maximum spectral radius among all n-vertex graphs not containing F, then G is a member of Ex(n,F) for n large enough. Then Cioabă, Desai and Tait's conjecture is completely solved. Furthermore, we give a stronger result.

The hat guessing number of graphs

Consider the following hat guessing game: n players are placed on n vertices of a graph, each wearing a hat whose color is arbitrarily chosen from a set of q possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. Given a graph G, its hat guessing number HG(G) is the largest integer q such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of q possible colors.In 2008, Butler et al. asked whether the hat guessing number of the complete bipartite graph Kn,n is at least some fixed positive (fractional) power of n. We answer this question affirmatively, showing that for sufficiently large n, the complete r-partite graph Kn,…,n satisfies HG(Kn,…,n)=Ω(nr−1r−o(1)). Our guessing strategy is based on a probabilistic construction and other combinatorial ideas, and can be extended to show that HG(C→n,…,n)=Ω(n1r−o(1)), where C→n,…,n is the blow-up of a directed r-cycle, and where for directed graphs each player sees only the hat colors of his outneighbors.Additionally, we consider related problems like the relation between the hat guessing number and other graph parameters, and the linear hat guessing number, where the players are only allowed to use affine linear guessing strategies. Several nonexistence results are obtained by using well-known combinatorial tools, including the Lovász Local Lemma and the Combinatorial Nullstellensatz. Among other results, it is shown that under certain conditions, the linear hat guessing number of Kn,n is at most 3, exhibiting a huge gap from the Ω(n12−o(1)) (nonlinear) hat guessing number of this graph.

The change of distance energy of some special complete multipartite graphs due to edge deletion

Let Kp1,p2,…,pr be a complete r-partite graph with r≥2 and pi≥2. Varghese et al. (2018) [14] conjectured that the distance energy of Kp1,p2,…,pr is always increased when an edge is deleted. In this paper, we prove that it is true for Kp,p,…,p and the tripartite Turán graph T(n,3). Subsequently, it is proved that, for any positive integer r, there exists a class of graphs that have r positive eigenvalues, but the distance energy of these graphs is always increased when an edge is deleted.

Extremal Matching Energy and the Largest Matching Root of Complete Multipartite Graphs

The matching energy ME(G) of a graph G was introduced by Gutman and Wagner, which is defined as the sum of the absolute values of the roots of the matching polynomial m(G, x). The largest matching root λ1(G) is the largest root of the matching polynomial m(G, x). Let denote the complete r‐partite graph with order n = n1 + n2 + …+nr, where r &gt; 1. In this paper, we prove that, for the given values n and r, both the matching energy ME(G) and the largest matching root λ1(G) of complete r‐partite graphs are minimal for complete split graph CS(n, r − 1) and are maximal for Turán graph T(n, r).

Sparse multipartite graphs as partition universal for graphs with bounded degree

For graphs G and H, let $$G\rightarrow (H,H)$$ signify that any red/blue edge coloring of G contains a monochromatic H as a subgraph. Denote $$\mathcal {H}(\Delta ,n)=\{H:|V(H)|=n,\Delta (H)\le \Delta \}$$ . For any $$\Delta $$ and n, we say that G is partition universal for $$\mathcal {H}(\Delta ,n)$$ if $$G\rightarrow (H,H)$$ for every $$H\in \mathcal {H}(\Delta ,n)$$ . Let $$G_r(N,p)$$ be the random spanning subgraph of the complete r-partite graph $$K_r(N)$$ with N vertices in each part, in which each edge of $$K_r(N)$$ appears with probability p independently and randomly. We prove that for fixed $$\Delta \ge 2$$ there exist constants r, B and C depending only on $$\Delta $$ such that if $$N\ge Bn$$ and $$p=C(\log N/N)^{1/\Delta }$$ , then asymptotically almost surely $$G_r(N,p)$$ is partition universal for $$\mathcal {H}(\Delta ,n)$$ .

On the Unit Graph of a Non-commutative Ring

Let R be a ring with non-zero identity. The unit graph G(R) of R is a graph with elements of R as its vertices and two distinct vertices a and b are adjacent if and only if a + b is a unit element of R. It was proved that if R is a commutative ring and 𝔪 is a maximal ideal of R such that |R/𝔪| = 2, then G(R) is a complete bipartite graph if and only if (R, 𝔪) is a local ring. In this paper we generalize this result by showing that if R is a ring (not necessarily commutative), then G(R) is a complete r-partite graph if and only if (R, 𝔪) is a local ring and r = |R/𝔪| = 2n for some n ∈ ℕ or R is a finite field. Among other results we show that if R is a left Artinian ring, 2 ∈ U(R) and the clique number of G(R) is finite, then R is a finite ring.

Equitable colorings of Cartesian products with balanced complete multipartite graphs

A proper vertex coloring of a graph is equitable if the sizes of any two color classes differ by at most one. The equitable chromatic number of a graph G, denoted by χ=(G), is the minimum k such that G is equitably k-colorable. Lin and Chang conjectured that for any (nontrivial) connected graphs G and H, χ=(G□H)≤χ(G)χ(H), where □ denotes the Cartesian product. In this paper, we prove the conjecture when G or H is a balanced complete multipartite graph. More precisely, we show a stronger result that for any graph H with χ(H)≥2, χ=(Kr(n)□H)≤r⌈χ(H)−1r−1⌉, where r≥2, n≥1 and Kr(n) denotes the balanced complete r-partite graph with part size n.

Binomial Edge Ideals of Complete Multipartite Graphs

In this article, we prove some results about the binomial edge ideal J G of a complete r-partite graph G = K a 1,…, a r : (1) characterization of unmixedness of J G and Cohen–Macaulayness of the residue ring S/J G , (2) F-purity of S/J G , and (3) the equality of the symbolic and the ordinary powers of J G .

On a generalization of Rubin's theorem

The work is devoted to the calculation of asymptotic value of the choice number of the complete r-partite graph Km* r = Km, …, m with equal part size m. We obtained the asymptotics in the case lnr = o(lnm). The proof generalizes the classical result of A.L. Rubin for the case r = 2. © 2010 Wiley Periodicals, Inc. J Graph Theory 67: 226–234, 2011 67: 226-234, 2011 © 2011 Wiley Periodicals, Inc.

Simple Graphs and Zero-divisor Semigroups

In this paper, we provide examples of graphs which uniquely determine a zero-divisor semigroup. We show two classes of graphs that have no corresponding semigroups. Especially, we prove that no complete r-partite graph together with two or more end vertices (each linked to distinct vertices) has corresponding semigroups.

Graphs with many r -cliques have large complete r -partite subgraphs

We prove that for all $r\geq2$ and c>0, every graph of order n with at least cn^{r} cliques of order r contains a complete r-partite graph with each part of size $\lfloor c^{r}\log n \rfloor.$ This result implies a concise form of the Erd\H{o}s-Stone theorem.

The intersection number of complete r-partite graphs

Latin squares C, D of order n are called pseudo-orthogonal if any two rows of the matrices C and D have exactly one common element. We give conditions for existence of families consisting of t pseudo-orthogonal Latin squares of order n. It is proved that the intersection number of a complete r-partite graph equals n2 if and only if there exists a family consisting of r – 2 pairwise pseudo-orthogonal Latin squares of order n. It is proved that if , where prols(n) is the maximum t such that there exists a set of t pseudo-orthogonal Latin squares of order n, then the intersection number of the graph is equal to n2. Applications of the obtained results to calculating the intersection number of some graphs are given.

Eigenvalues and forbidden subgraphs I

Suppose a graph G have n vertices, m edges, and t triangles. Letting λ n ( G) be the largest eigenvalue of the Laplacian of G and μ n ( G) be the smallest eigenvalue of its adjacency matrix, we prove that λ n ( G ) ⩾ 2 m 2 - 3 nt m ( n 2 - 2 m ) n , μ n ( G ) ⩽ 3 n 3 t - 4 m 3 nm ( n 2 - 2 m ) , with equality if and only if G is a regular complete multipartite graph. Moreover, if G is K r+1 -free, then λ n ( G ) ⩾ 2 mn ( r - 1 ) ( n 2 - 2 m ) with equality if and only if G is a regular complete r-partite graph.

A Note on Cyclic m-cycle Systems of Kr(m)

It was proved by Buratti and Del Fra that for each pair of odd integers r and m, there exists a cyclic m-cycle system of the balanced complete r-partite graph Kr(m) except for the case when r=m=3. In this note, we study the existence of a cyclic m-cycle system of Kr(m) where r or m is even. Combining the work of Buratti and Del Fra, we prove that cyclic m-cycle systems of Kr(m) exist if and only if (a) Kr(m) is an even graph (b) (r, m)≠ (3, 3) and (c) (r,m)≢ (t , 2) (mod 4) where t ∈ {2,3}.

When a zero-divisor graph is planar or a complete r-partite graph

Let Γ( R) be the zero-divisor graph of a commutative ring R. An interesting question was proposed by Anderson, Frazier, Lauve, and Livingston: For which finite commutative rings R is Γ( R) planar? We give an answer to this question. More precisely, we prove that if R is a local ring with at least 33 elements, and Γ( R)≠∅, then Γ( R) is not planar. We use the set of the associated primes to find the minimal length of a cycle in Γ( R). Also, we determine the rings whose zero-divisor graphs are complete r-partite graphs and show that for any ring R and prime number p, p⩾3, if Γ( R) is a finite complete p-partite graph, then | Z( R)|= p 2, | R|= p 3, and R is isomorphic to exactly one of the rings Z p 3 , Z p[x,y] (xy,y 2−x) , Z p 2 [y] (py,y 2−ps) , where 1⩽ s< p.

Steiner graphical t-wise balanced designs of type nr

In this article we completely determines all proper Steiner graphical t-wise balanced designs whose automorphisms contain the group SnwrSr (wreath product) acting on the set of n2r2 edges of the complete r-partite graph, Kn,n,…,n.

On the choosability of complete multipartite graphs with part size three

Let K m∗r be the complete r-partite graph with m vertices in each part. Erdős, Rubin, and Taylor showed that K 2∗r is r-choosable and suggested the problem of determining the choosability of K m∗r . We show that K 3∗r is exactly ⌈(4 r−1)/3⌉ choosable.