Acyclic edge coloring of subcubic graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and it is denoted by a^′(G). From a result of Burnstein it follows that all subcubic graphs are acyclically edge colorable using five colors. This result is tight since there are 3-regular graphs which require five colors. In this paper we prove that any non-regular connected graph of maximum degree 3 is acyclically edge colorable using at most four colors. This result is tight since all edge maximal non-regular connected graphs of maximum degree 3 require four colors.     

On Rauzy graph sequences of infinite words

Under study are the sequences of Rauzy graphs (i.e., the graphs of subwords overlapping) of infinite words. The k-stretching of a graph is the graph we obtain by replacing each edge with a chain of length k. Considering a sequence of strongly connected directed graphs of maximal in and out vertex degrees equal to s, we prove that it is, up to stretchings, a subsequence of a Rauzy graphs sequence of some uniformly recurrent infinite word on s-letter alphabet. The language of a word of this kind and stretching for a given sequence of graphs are constructed explicitly.     

On perturbations of almost distance-regular graphs

In this paper we show that certain almost distance-regular graphs, the so-called h-punctually walk-regular graphs, can be characterized through the cospectrality of their perturbed graphs. A graph G with diameter D is called h-punctually walk-regular, for a given h≤D, if the number of paths of length ? between a pair of vertices u,v at distance h depends only on ?. The graph perturbations considered here are deleting a vertex, adding a loop, adding a pendant edge, adding/removing an edge, amalgamating vertices, and adding a bridging vertex. We show that for walk-regular graphs some of these operations are equivalent, in the sense that one perturbation produces cospectral graphs if and only if the others do. Our study is based on the theory of graph perturbations developed by Cvetkovi?, Godsil, McKay, Rowlinson, Schwenk, and others. As a consequence, some new characterizations of distance-regular graphs are obtained.     

Super restricted edge connectivity of regular graphs with two orbits

It is well known that the edge-connectivity of a simple, connected, vertex transitive graph attains its regular degree. It is then natural to consider the relationship between the graph’s edge connectivity and the number of orbits of its automorphism group. In [6], Liu and Meng (2008) studied the edge connectivity of regular double-orbits graphs. Later, Lin et al. (in press) [10] characterized the λ′-optimal 3-regular double-orbit graph and given a sufficient condition for the k-regular double-orbit graphs to be optimal. In this note, we characterize the super restricted edge connected k-regular double-orbit graphs with grith at least 6.     

Connected Q-integral graphs with maximum edge-degree less than or equal to 8

Graphs with integral signless Laplacian spectrum are called Q-integral graphs. The number of adjacent edges to an edge is defined as the edge-degree of that edge. The Q-spectral radius of a graph is the largest eigenvalue of its signless Laplacian. In 2019, Park and Sano [16] studied connected Q-integral graphs with the maximum edge-degree at most six. In this article, we extend their result and study the connected Q-integral graphs with maximum edge-degree less than or equal to eight. Further, we give an upper bound and a lower bound for the maximum edge-degree of a connected Q-integral graph with respect to its Q-spectral radius. As a corollary, we show that the Q-spectral radius of the connected edge-non-regular Q-integral graph with maximum edge-degree five is six, which we anticipate to be a key for solving the unsolved problem of characterizing such graphs.     

Generating 3-vertex connected spanning subgraphs

In this paper we present an algorithm to generate all minimal 3-vertex connected spanning subgraphs of an undirected graph with n vertices and m edges in incremental polynomial time, i.e., for every K we can generate K (or all) minimal 3-vertex connected spanning subgraphs of a given graph in O(K²log(K)m²+K²m³) time, where n and m are the number of vertices and edges of the input graph, respectively. This is an improvement over what was previously available and is the same as the best known running time for generating 2-vertex connected spanning subgraphs. Our result is obtained by applying the decomposition theory of 2-vertex connected graphs to the graphs obtained from minimal 3-vertex connected graphs by removing a single edge.     

Equivalences and the complete hierarchy of intersection graphs of paths in a tree

An (h,s,t)-representation of a graph G consists of a collection of subtrees of a tree T, where each subtree corresponds to a vertex in G, such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at most s, (iii) there is an edge between two vertices in the graph G if and only if the corresponding subtrees have at least t vertices in common in T. The class of graphs that have an (h,s,t)-representation is denoted by [h,s,t]. It is well known that the class of chordal graphs corresponds to the class [3, 3, 1]. Moreover, it was proved by Jamison and Mulder that chordal graphs correspond to orthodox-[3, 3, 1] graphs defined below.In this paper, we investigate the class of [h,2,t] graphs, i.e., the intersection graphs of paths in a tree. The [h,2,1] graphs are also known as path graphs [F. Gavril, A recognition algorithm for the intersection graphs of paths in trees, Discrete Math. 23 (1978) 211-227] or VPT graphs [M.C. Golumbic, R.E. Jamison, Edge and vertex intersection of paths in a tree, Discrete Math. 55 (1985) 151-159], and [h,2,2] graphs are known as the EPT graphs. We consider variations of [h,2,t] by three main parameters: h, t and whether the graph has an orthodox representation. We give the complete hierarchy of relationships between the classes of weakly chordal, chordal, [h,2,t] and orthodox-[h,2,t] graphs for varied values of h and t.     

The harmonious coloring problem is NP-complete for interval and permutation graphs

In this paper, we prove that the harmonious coloring problem is NP-complete for connected interval and permutation graphs. Given a simple graph G, a harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number is the least integer k for which G admits a harmonious coloring with k colors. Extending previous work on the NP-completeness of the harmonious coloring problem when restricted to the class of disconnected graphs which are simultaneously cographs and interval graphs, we prove that the problem is also NP-complete for connected interval and permutation graphs.     

On claw-free t-perfect graphs

A graph is called t-perfect, if its stable set polytope is defined by non-negativity, edge and odd-cycle inequalities. We characterise the class of all claw-free t-perfect graphs by forbidden t-minors, and show that they are 3-colourable. Moreover, we determine the chromatic number of claw-free h-perfect graphs and give a polynomial-time algorithm to compute an optimal colouring.     

A class of h-perfect graphs

A graph is said to be h-perfect if the convex hull of its independent sets is defined by the constraints corresponding to cliques and odd holes, and the nonnegativity constraints. Series-parallel graphs and perfect graphs are h-perfect. The purpose of this paper is to extend the class of graphs known to be h-perfect. Thus, given a graph which is the union of a bipartite graph G₁ and a graph G₂ having exactly two common nodes a and b, and no edge in common, we prove that G is h-perfect if so is the graph obtained from G by replacing G₁ by an a-b chain (the length of which depends on G₁). This result enables us to prove that the graph obtained by substituting bipartite graphs for edges of a series-parallel graph is h-perfect, and also that the identification of two nodes of a bipartite graph yields an h-perfect graph (modulo a reduction which preserves h-perfection).     

Note on Halin's theorem on minimally connected graphs

In this note it is shown that any finite directed graph of strong connectivity n contains either a vertex with indegree n, a vertex with outdegree n, or an edge whose removal does not decrease the connectivity. This is a directed graph counterpart of Halin's theorem on undirected graphs. It is pointed out that only a few preparations and modifications are necessary to make his proof valid for directed graphs.     

A characterization of partial directed line graphs

Can a directed graph be completed to a directed line graph? If possible, how many arcs must be added? In this paper we address the above questions characterizing partial directed line (PDL) graphs, i.e., partial subgraph of directed line graphs. We show that for such class of graphs a forbidden configuration criterion and a Krausz's like theorem are equivalent characterizations. Furthermore, the latter leads to a recognition algorithm that requires O(m) worst case time, where m is the number of arcs in the graph. Given a partial line digraph, our characterization allows us to find a minimum completion to a directed line graph within the same time bound.The class of PDL graphs properly contains the class of directed line graphs, characterized in [J. Blazewicz, A. Hertz, D. Kobler, D. de Werra, On some properties of DNA graphs, Discrete Appl. Math. 98(1-2) (1999) 1-19], hence our results generalize those already known for directed line graphs. In the undirected case, we show that finding a minimum line graph edge completion is NP-hard, while the problem of deciding whether or not an undirected graph is a partial graph of a simple line graph is trivial.     

The bondage numbers of graphs with small crossing numbers

The bondage number b(G) of a nonempty graph G is the cardinality of a smallest edge set whose removal from G results in a graph with domination number greater than the domination number γ(G) of G. Kang and Yuan proved b(G)?8 for every connected planar graph G. Fischermann, Rautenbach and Volkmann obtained some further results for connected planar graphs. In this paper, we generalize their results to connected graphs with small crossing numbers.     

Graph classes and the complexity of the graph orientation minimizing the maximum weighted outdegree

Given an undirected graph with edge weights, we are asked to find an orientation, that is, an assignment of a direction to each edge, so as to minimize the weighted maximum outdegree in the resulted directed graph. The problem is called MMO, and is a restricted variant of the well-known minimum makespan problem. As in previous studies, it is shown that MMO is in P for trees, weak NP-hard for planar bipartite graphs, and strong NP-hard for general graphs. There are still gaps between those graph classes. The objective of this paper is to show tighter thresholds of complexity: We show that MMO is (i) in P for cactus graphs, (ii) weakly NP-hard for outerplanar graphs, and also (iii) strongly NP-hard for graphs which are both planar and bipartite. This implies the NP-hardness for P₄-bipartite, diamond-free or house-free graphs, each of which is a superclass of cactus. We also show (iv) the NP-hardness for series-parallel graphs and multi-outerplanar graphs, and (v) present a pseudo-polynomial time algorithm for graphs with bounded treewidth.     

Theory of path length distributions I

The Path Length Distribution (PLD) of a (p, q) graph is defined to be the array (X₀, X₁, X₂, …, X_p-1), where X₀ is the number of (unordered) pairs of vertices which have no path connecting them and X_l, 1 ≦ l ≦ p-1, is the number of pairs of vertices which are connected by a path of length l (see [1, 2]). The topic of this paper is the occurence of non-isomorphic graphs having the same path length distribution. For trees, a constructive procedure is given, showing that for any positive integer N there exist N non-isomorphic trees of diameter four which have the same PLD. Also considered are PLD-maximal graphs — those graphs with p vertices such that all pairs of vertices are connected by a path of length l for 2 ≦ l ≦ p-1. In addition to providing more examples of non-isomorphic graphs having the same PLD, PLD-maximal graphs are of intrinsic interest. For PLD-maximal graphs, we give sufficient degree and edge conditions and a necessary edge condition.     

Rainbow Connection and Graph Products

A path in an edge colored graph G is called a rainbow path if all its edges have pairwise different colors. Then G is rainbow connected if there exists a rainbow path between every pair of vertices of G and the least number of colors needed to obtain a rainbow connected graph is the rainbow connection number. If we demand that there must exist a shortest rainbow path between every pair of vertices, we speak about strongly rainbow connected graph and the strong rainbow connection number. In this paper we study the (strong) rainbow connection number on the direct, strong, and lexicographic product and present several upper bounds for these products that are attained by many graphs. Several exact results are also obtained.     

Generating unlabeled connected cubic planar graphs uniformly at random

We present an expected polynomial time algorithm to generate an unlabeled connected cubic planar graph uniformly at random. We first consider rooted connected cubic planar graphs, i.e., we count connected cubic planar graphs up to isomorphisms that fix a certain directed edge. Based on decompositions along the connectivity structure, we derive recurrence formulas for the exact number of rooted cubic planar graphs. This leads to rooted 3‐connected cubic planar graphs, which have a unique embedding on the sphere. Special care has to be taken for rooted graphs that have a sense‐reversing automorphism. Therefore we introduce the concept of colored networks, which stand in bijective correspondence to rooted 3‐connected cubic planar graphs with given symmetries. Colored networks can again be decomposed along the connectivity structure. For rooted 3‐connected cubic planar graphs embedded in the plane, we switch to the dual and count rooted triangulations. Since all these numbers can be evaluated in polynomial time using dynamic programming, rooted connected cubic planar graphs can be generated uniformly at random in polynomial time by inverting the decomposition along the connectivity structure. To generate connected cubic planar graphs without a root uniformly at random, we apply rejection sampling and obtain an expected polynomial time algorithm. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Uniformly finite-to-one and onto extensions of homomorphisms between strongly connected graphs

For a homomorphism between directed graphs G₁ and G₂, its extension is the mapping of the set of all paths in G₁ into the set of all paths in G₂ obtained by naturally extending it. We investigate the properties of uniformly finite-to-one and onto extensions of homomorphisms of directed graphs, essentially the properties of uniformly finite-to-one and onto extensions of homomorphisms between strongly connected directed graphs. We also describe applications of our results on homomorphisms of directed graphs to the theory of a class of symbolic flows called subshifts of finite type.     

Pseudograph associahedra

Given a simple graph G, the graph associahedron KG is a simple polytope whose face poset is based on the connected subgraphs of G. This paper defines and constructs graph associahedra in a general context, for pseudographs with loops and multiple edges, which are also allowed to be disconnected. We then consider deformations of pseudograph associahedra as their underlying graphs are altered by edge contractions and edge deletions.     

Strong structural properties of unidirectional star graphs

Day and Tripathi [K. Day, A. Tripathi, Unidirectional star graphs, Inform. Process. Lett. 45 (1993) 123-129] proposed an assignment of directions on the star graphs and derived attractive properties for the resulting directed graphs: an important one is that they are strongly connected. In [E. Cheng, M.J. Lipman, On the Day-Tripathi orientation of the star graphs: Connectivity, Inform. Process. Lett. 73 (2000) 5-10] it is shown that the Day-Tripathi orientations are in fact maximally arc-connected when n is odd; when n is even, they can be augmented to maximally arc-connected digraphs by adding a minimum set of arcs. This gives strong evidence that the Day-Tripathi orientations are good orientations. In [E. Cheng, M.J. Lipman, Connectivity properties of unidirectional star graphs, Congr. Numer. 150 (2001) 33-42] it is shown that vertex-connectivity is maximal, and that if we delete as many vertices as the connectivity, we can create at most two strong connected components, at most one of which is not a singleton. In this paper we prove an asymptotically sharp upper bound for the number of vertices we can delete without creating two nonsingleton strong components, and we also give sharp upper bounds on the number of singletons that we might create.