Independence equivalence classes of cycles

The independence equivalence class of a graph G is the set of graphs that have the same independence polynomial as G. Beaton, Brown and Cameron (2019) found the independence equivalence classes of even cycles, and raised the problem of finding the independence equivalence class of odd cycles. The problem is completely solved in this paper.     

A study of 3-arc graphs

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple (v,u,x,y) of vertices such that both (v,u,x) and (u,x,y) are paths of length two. The 3-arc graph of a graph G is defined to have the arcs of G as vertices such that two arcs uv,xy are adjacent if and only if (v,u,x,y) is a 3-arc of G. In this paper, we study the independence, domination and chromatic numbers of 3-arc graphs and obtain sharp lower and upper bounds for them. We introduce a new notion of arc-coloring of a graph in studying vertex-colorings of 3-arc graphs.     

An Optimal Algorithm for the <Emphasis Type="Italic">k</Emphasis>-Fixed-Endpoint Path Cover on Proper Interval Graphs

In this paper we consider the k-fixed-endpoint path cover problem on proper interval graphs, which is a generalization of the path cover problem. Given a graph G and a set T of k vertices, a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint simple paths that covers the vertices of G, such that the vertices of T are all endpoints of these paths. The goal is to compute a k-fixed-endpoint path cover of G with minimum cardinality. We propose an optimal algorithm for this problem with runtime O(n), where n is the number of intervals in G. This algorithm is based on the Stair Normal Interval Representation (SNIR) matrix that characterizes proper interval graphs. In this characterization, every maximal clique of the graph is represented by one matrix element; the proposed algorithm uses this structural property, in order to determine directly the paths in an optimal solution.     

Covering the vertices of a graph with cycles of bounded length

Let c_k(G) be the minimum number of elementary cycles of length at most k necessary to cover the vertices of a given graph G. In this work, we bound c_k(G) by a function of the order of G and its independence number.     

An application of Vizing and Vizing-like adjacency lemmas to Vizing’s Independence Number Conjecture of edge chromatic critical graphs

In 1968, Vizing conjectured that, if G is a Δ-critical graph with n vertices, then , where α(G) is the independence number of G. In this note, we apply Vizing and Vizing-like adjacency lemmas to this problem and obtain better bounds for Δ∈{7,…,19}.     

A note on Vizing's independence number conjecture of edge chromatic critical graphs

In 1968, Vizing conjectured that, if G is a Δ-critical graph with n vertices, then α(G)?n/2, where α(G) is the independence number of G. In this note, we verify this conjecture for n?2Δ.     

The independence polynomial of rooted products of graphs

A stable (or independent) set in a graph is a set of pairwise nonadjacent vertices thereof. The stability numberα(G) is the maximum size of stable sets in a graph G. The independence polynomial of G is

     

Well-covered graphs and factors

A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α. Plummer [Some covering concepts in graphs, J. Combin. Theory 8 (1970) 91-98] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. Every well-covered graph G without isolated vertices has a perfect [1,2]-factor F_G, i.e. a spanning subgraph such that each component is 1-regular or 2-regular. Here, we characterize all well-covered graphs G satisfying α(G)=α(F_G) for some perfect [1,2]-factor F_G. This class contains all well-covered graphs G without isolated vertices of order n with α?(n-1)/2, and in particular all very well-covered graphs.     

Partitioning 2-edge-colored graphs by monochromatic paths and cycles

We present results on partitioning the vertices of 2-edge-colored graphs into monochromatic paths and cycles. We prove asymptotically the two-color case of a conjecture of Sárközy: the vertex set of every 2-edge-colored graph can be partitioned into at most 2α(G) monochromatic cycles, where α(G) denotes the independence number of G. Another direction, emerged recently from a conjecture of Schelp, is to consider colorings of graphs with given minimum degree. We prove that apart from o(|V (G)|) vertices, the vertex set of any 2-edge-colored graph G with minimum degree at least \(\tfrac{{(1 + \varepsilon )3|V(G)|}} {4}\) can be covered by the vertices of two vertex disjoint monochromatic cycles of distinct colors. Finally, under the assumption that \(\bar G\) does not contain a fixed bipartite graph H, we show that in every 2-edge-coloring of G, |V (G)| ? c(H) vertices can be covered by two vertex disjoint paths of different colors, where c(H) is a constant depending only on H. In particular, we prove that c(C ₄)=1, which is best possible.     

On the Soluble Graph of a Finite Simple Group

The maximal independent sets of the soluble graph of a finite simple group G are studied and their independence number is determined. In particular, it is shown that this graph in many cases has an independent set with three vertices.     

A basic elementary extension of the Duchet-Meyniel theorem

The Conjecture of Hadwiger implies that the Hadwiger number h times the independence number α of a graph is at least the number of vertices n of the graph. In 1982 Duchet and Meyniel [P. Duchet, H. Meyniel, On Hadwiger’s number and the stability number, Ann. of Discrete Math. 13 (1982) 71-74] proved a weak version of the inequality, replacing the independence number α by 2α−1, that is,

(2α−1)⋅h≥n.     

Connected graphs with a minimal number of spanning trees

The problem studied is the following: Find a simple connected graph G with given numbers of vertices and edges which minimizes the number t_μ(G), the number of spanning trees of the multigraph obtained from G by adding μ parallel edges between every pair of distinct vertices. If G is nearly complete (the number of edges qis ≥(₂^P)?p+2 where p is the number of vertices), then the solution to the minimization problem is unique (up to isomorphism) and the same for all values of μ. The present paper investigates the case whereq<(₂^P)?p+2. In this case the solution is not always unique and there does not always exist a common solution for all values of μ. A (small) class of graphs is given such that for any μ there exists a solution to the problem which is contained in this class. For μ = 0 there is only one graph in the class which solves the problem. This graph is described and the minimum value of t₀(G) is found. In order to derive these results a representation theorem is proved for the cofactors of a special class of matrices which contains the tree matrices associated with graphs.     

Finding a minimum path cover of a distance-hereditary graph in polynomial time

A path cover of a graph G=(V,E) is a set of pairwise vertex-disjoint paths such that the disjoint union of the vertices of these paths equals the vertex set V of G. The path cover problem is, given a graph, to find a path cover having the minimum number of paths. The path cover problem contains the Hamiltonian path problem as a special case since finding a path cover, consisting of a single path, corresponds directly to the Hamiltonian path problem. A graph is a distance-hereditary graph if each pair of vertices is equidistant in every connected induced subgraph containing them. The complexity of the path cover problem on distance-hereditary graphs has remained unknown. In this paper, we propose the first polynomial-time algorithm, which runs in O(|V|⁹) time, to solve the path cover problem on distance-hereditary graphs.     

Rainbow Connection Number and Independence Number of a Graph

A path in an edge-colored graph is called rainbow if any two edges of the path have distinct colors. An edge-colored graph is called rainbow connected if there exists a rainbow path between every two vertices of the graph. For a connected graph G, the minimum number of colors that are needed to make G rainbow connected is called the rainbow connection number of G, denoted by rc(G). In this paper, we investigate the relation between the rainbow connection number and the independence number of a graph. We show that if G is a connected graph without pendant vertices, then \(\mathrm{rc}(G)\le 2\alpha (G)-1\). An example is given showing that the upper bound \(2\alpha (G)-1\) is equal to the diameter of G, and so the upper bound is sharp since the diameter of G is a lower bound of \(\mathrm{rc}(G)\).     

On the Estrada and Laplacian Estrada indices of graphs

The Estrada index of a graph G is defined as , where λ₁,λ₂,…,λ_n are the eigenvalues of G. The Laplacian Estrada index of a graph G is defined as , where μ₁,μ₂,…,μ_n are the Laplacian eigenvalues of G. An edge grafting operation on a graph moves a pendent edge between two pendent paths. We study the change of Estrada index of graph under edge grafting operation between two pendent paths at two adjacent vertices. As the application, we give the result on the change of Laplacian Estrada index of bipartite graph under edge grafting operation between two pendent paths at the same vertex. We also determine the unique tree with minimum Laplacian Estrada index among the set of trees with given maximum degree, and the unique trees with maximum Laplacian Estrada indices among the set of trees with given diameter, number of pendent vertices, matching number, independence number and domination number, respectively.     

Independence polynomials of circulants with an application to music

The independence polynomial of a graph G is the generating function I(G,x)=∑_k≥0i_kx^k, where i_k is the number of independent sets of cardinality k in G. We show that the problem of evaluating the independence polynomial of a graph at any fixed non-zero number is intractable, even when restricted to circulants. We provide a formula for the independence polynomial of a certain family of circulants, and its complement. As an application, we derive a formula for the number of chords in an n-tet musical system (one where the ratio of frequencies in a semitone is 2^1/n) without ‘close’ pitch classes.     

Hall ratio of the Mycielski graphs

Let n(G) denote the number of vertices of a graph G and let α(G) be the independence number of G, the maximum number of pairwise nonadjacent vertices of G. The Hall ratio of a graph G is defined by

     

Asymptotic values of the Hall-ratio for graph powers

The Hall-ratio of a graph G is the ratio of the number of vertices and the independence number maximized over all subgraphs of G. We investigate asymptotic values of the Hall-ratio with respect to different graph powers.     

Linear Coloring of Planar Graphs Without 4-Cycles

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of G is the smallest number of colors in a linear coloring of G. In this paper, we prove that if G is a planar graph without 4-cycles, then lc ${(G)\le \lceil \frac {\Delta}2\rceil+8}$ , where Δ denotes the maximum degree of G.     

Domination Defect in Graphs: Guarding With Fewer Guards

In this paper, we introduce a new graph parameter called the domination defect of a graph. The domination number γ of a graph G is the minimum number of vertices required to dominate the vertices of G. Due to the minimality of γ, if a set of vertices of G has cardinality less than γ then there are vertices of G that are not dominated by that set. The k-domination defect of G is the minimum number of vertices which are left un-dominated by a subset of γ - k vertices of G. We study different bounds on the k-domination defect of a graph G with respect to the domination number, order, degree sequence, graph homomorphisms and the existence of efficient dominating sets. We also characterize the graphs whose domination defect is 1 and find exact values of the domination defect for some particular classes of graphs.