On low degree k-ordered graphs

A simple graph G is k-ordered (respectively, k-ordered hamiltonian) if, for any sequence of k distinct vertices v₁,…,v_k of G, there exists a cycle (respectively, a hamiltonian cycle) in G containing these k vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability, and motivated by the fact that k-orderedness of a graph implies (k-1)-connectivity, they posed the question of the existence of low degree k-ordered hamiltonian graphs. We construct an infinite family of graphs, which we call bracelet graphs, that are (k-1)-regular and are k-ordered hamiltonian for odd k. This result provides the best possible answer to the question of the existence of low degree k-ordered hamiltonian graphs for odd k. We further show that for even k, there exist no k-ordered bracelet graphs with minimum degree k-1 and maximum degree less than k+2, and we exhibit an infinite family of bracelet graphs with minimum degree k-1 and maximum degree k+2 that are k-ordered for even k. A concept related to k-orderedness, namely that of k-edge-orderedness, is likewise strongly related to connectivity properties. We study this relation and give bounds on the connectivity necessary to imply k-(edge-)orderedness properties.     

On 4-connected 4-regular graphs without even cycle decompositions

An even cycle decomposition of a graph is a partition of its edges into even cycles. Markström constructed infinitely many 2-connected 4-regular graphs without even cycle decompositions. Má?ajová and Mazák then constructed an infinite family of 3-connected 4-regular graphs without even cycle decompositions. In this note, we further show that there exists an infinite family of 4-connected 4-regular graphs without even cycle decompositions.     

Maximum independent sets in 3- and 4-regular Hamiltonian graphs

Smooth 4-regular Hamiltonian graphs are generalizations of cycle-plus-triangles graphs. While the latter have been shown to be 3-choosable, 3-colorability of the former is NP-complete. In this paper we first show that the independent set problem for 3-regular Hamiltonian planar graphs is NP-complete, and using this result we show that this problem is also NP-complete for smooth 4-regular Hamiltonian graphs. We also show that this problem remains NP-complete if we restrict the problem to the existence of large independent sets (i.e., independent sets whose size is at least one third of the order of the graphs).     

Clique-transversal sets in 4-regular claw-free graphs

A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted by τ _c (G), is the minimum cardinality of a clique-transversal set in G. In this paper we give the exact value of the clique-transversal number for the line graph of a complete graph. Also, we give a lower bound on the clique-transversal number for 4-regular claw-free graphs and characterize the extremal graphs achieving the lower bound.     

Construction for bicritical graphs and k-extendable bipartite graphs

A graph G is said to be bicritical if G-u-v has a perfect matching for every choice of a pair of points u and v. Bicritical graphs play a central role in decomposition theory of elementary graphs with respect to perfect matchings. As Plummer pointed out many times, the structure of bicritical graphs is far from completely understood. This paper presents a concise structure characterization on bicritical graphs in terms of factor-critical graphs and transversals of hypergraphs. A connected graph G with at least 2k+2 points is said to be k-extendable if it contains a matching of k lines and every such matching is contained in a perfect matching. A structure characterization for k-extendable bipartite graphs is given in a recursive way. Furthermore, this paper presents an O(mn) algorithm for determining the extendability of a bipartite graph G, the maximum integer k such that G is k-extendable, where n is the number of points and m is the number of lines in G.     

On extremal k-outconnected graphs

Let G be a minimally k-connected graph with n nodes and m edges. Mader proved that if n?3k-2 then m?k(n-k), and for n?3k-1 an equality is possible if, and only if, G is the complete bipartite graph K_k,n-k. Cai proved that if n?3k-2 then m?⌊(n+k)²/8⌋, and listed the cases when this bound is tight.In this paper we prove a more general theorem, which implies similar results for minimally k-outconnected graphs; a graph is called k-outconnected from r if it contains k internally disjoint paths from r to every other node.     

On k-regular functions with values in a universal Clifford algebra

In this paper, we mainly study properties of nullsolutions of the operator Dk (k∈N^∗=N?{0}), so-called k-regular functions. Firstly, we study the set of all homogeneous polynomials of degree p in x₁,…,xn which are k-regular in the whole Rn, clearly is a right module over C(Vn,n), we construct a basis for the right module . Secondly, we study the k-regular and analytic functions, and we give the Taylor expansions for these functions. At last, the corresponding Taylor expansions for k-regular functions are given since each k-regular function is a real analytic function.     

On the Clique-Transversal Number in （Claw,K4）-Free 4-Regular Graphs

A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G.The clique-transversal number,denoted by τC(G),is the minimum cardinality of a clique-transversal set in G.In this paper,we first present a lower bound on τC(G) and characterize the extremal graphs achieving the lower bound for a connected(claw,K4)-free 4-regular graph G.Furthermore,we show that for any 2-connected(claw,K4)-free 4-regular graph G of order n,its clique-transversal number equals to [n/3].     

直径为3的3-正则简单平面图的完全刻画

通过对子图和围长的研究,完全刻画了直径为3的3-正则简单平面图,获得了这类图仅有的11个非同构图.     

On a class of graphs without 3-stars

M. Numata described edge regular graphs without 3-stars. Allμ-subgraphs of these graphs are regular of the same valency. We prove that a connected graph without 3-stars all of whoseμ- subgraphs are regular of valencyα > 0 is either a triangular graph, or the Shläfli graph, or the icosahedron graph.     

A characterization of maximal non-k-factor-critical graphs

A graph G of order p is k-factor-critical,where p and k are positive integers with the same parity, if the deletion of any set of k vertices results in a graph with a perfect matching. G is called maximal non-k-factor-critical if G is not k-factor-critical but G+e is k-factor-critical for every missing edge e∉E(G). A connected graph G with a perfect matching on 2n vertices is k-extendable, for 1?k?n-1, if for every matching M of size k in G there is a perfect matching in G containing all edges of M. G is called maximal non-k-extendable if G is not k-extendable but G+e is k-extendable for every missing edge e∉E(G) . A connected bipartite graph G with a bipartitioning set (X,Y) such that |X|=|Y|=n is maximal non-k-extendable bipartite if G is not k-extendable but G+xy is k-extendable for any edge xy∉E(G) with x∈X and y∈Y. A complete characterization of maximal non-k-factor-critical graphs, maximal non-k-extendable graphs and maximal non-k-extendable bipartite graphs is given.     

Total minus domination in k-partite graphs

A function f defined on the vertices of a graph G=(V,E),f:V→{-1,0,1} is a total minus dominating function (TMDF) if the sum of its values over any open neighborhood is at least one. The weight of a TMDF is the sum of its function values over all vertices. The total minus domination number, denoted by , of G is the minimum weight of a TMDF on G. In this paper, a sharp lower bound on of k-partite graphs is given.     

Constant tolerance intersection graphs of subtrees of a tree

A chordal graph is the intersection graph of a family of subtrees of a host tree. In this paper we generalize this. A graph G=(V,E) has an (h,s,t)-representation if there exists a host tree T of maximum degree at most h, and a family of subtrees {S_v}_v∈V of T, all of maximum degree at most s, such that uv∈E if and only if |S_u∩S_v|?t. For given h,s, and t, there exist infinitely many forbidden induced subgraphs for the class of (h,s,t)-graphs. On the other hand, for fixed h?s?3, every graph is an (h,s,t)-graph provided that we take t large enough. Under certain conditions representations of larger graphs can be obtained from those of smaller ones by amalgamation procedures. Other representability and non-representability results are presented as well.     

On friendly index sets of 2-regular graphs

Let G be a graph with vertex set V and edge set E, and let A be an abelian group. A labeling f:V→A induces an edge labeling f^∗:E→A defined by f^∗(xy)=f(x)+f(y). For i∈A, let v_f(i)=card{v∈V:f(v)=i} and e_f(i)=card{e∈E:f^∗(e)=i}. A labeling f is said to be A-friendly if |v_f(i)−v_f(j)|≤1 for all (i,j)∈A×A, and A-cordial if we also have |e_f(i)−e_f(j)|≤1 for all (i,j)∈A×A. When A=Z₂, the friendly index set of the graph G is defined as {|e_f(1)−e_f(0)|:the vertex labelingf is Z₂-friendly}. In this paper we completely determine the friendly index sets of 2-regular graphs. In particular, we show that a 2-regular graph of order n is cordial if and only if n?2 (mod 4).     

Modelling gateway placement in wireless networks: Geometric k-centres of unit disc graphs

Motivated by the gateway placement problem in wireless networks, we consider the geometric k-centre problem on unit disc graphs: given a set of points P in the plane, find a set F of k points in the plane that minimizes the maximum graph distance from any vertex in P to the nearest vertex in F in the unit disc graph induced by P∪F. We show that the vertex 1-centre provides a 7-approximation of the geometric 1-centre and that a vertex k-centre provides a 13-approximation of the geometric k-centre, resulting in an O(kn)-time 26-approximation algorithm. We describe O(n²m)-time and O(n³)-time algorithms, respectively, for finding exact and approximate geometric 1-centres, and an O(mn2k)-time algorithm for finding a geometric k-centre for any fixed k. We show that the problem is NP-hard when k is an arbitrary input parameter. Finally, we describe an O(n)-time algorithm for finding a geometric k-centre in one dimension.     

Some 3‐connected 4‐edge‐critical non‐Hamiltonian graphs

Let γ(G) be the domination number of graph G, thus a graph G is k‐edge‐critical if γ (G) = k, and for every nonadjacent pair of vertices u and υ, γ(G + uυ) = k?1. In Chapter 16 of the book “Domination in Graphs—Advanced Topics,” D. Sumner cites a conjecture of E. Wojcicka under the form “3‐connected 4‐critical graphs are Hamiltonian and perhaps, in general (i.e., for any k ≥ 4), (k?1)‐connected, k‐edge‐critical graphs are Hamiltonian.” In this paper, we prove that the conjecture is not true for k = 4 by constructing a class of 3‐connected 4‐edge‐critical non‐Hamiltonian graphs. © 2005 Wiley Periodicals, Inc.     

Perfect 3-colorings on 6-regular graphs of order 9

The concept of a perfect coloring, introduced by P. Delsarte, generalizes the concept of completely regular code. We study the perfect 3-colorings (also known as the equitable partitions into three parts) on 6-regular graphs of order 9. A perfect n-colorings of a graph is a partition of its vertex set. It splits vertices into n parts A1, A2,...,An such that for all i,j∈{1,2,...,n}, each vertex of Ai is adjacent to aij vertices of Aj. The matrix A =(aij)n×n is called quotient matrix or parameter matrix. In this article, we start by giving an algorithm to find all different types of 6-regular graphs of order 9. Then, we classify all the realizable parameter matrices of perfect 3-colorings on 6-regular graphs of order 9.     

On k-planar crossing numbers

The k-planar crossing number of a graph is the minimum number of crossings of its edges over all possible drawings of the graph in k planes. We propose algorithms and methods for k-planar drawings of general graphs together with lower bound techniques. We give exact results for the k-planar crossing number of K_2k+1,q, for k?2. We prove tight bounds for complete graphs. We also study the rectilinear k-planar crossing number.