Snarks are a historical class of cubic graphs with peculiar properties motivated by the Four-Color Theorem. In nearly 100 years of search, since its definition by Peter Guthrie Tait in 1880, only five such graphs were identified which motivated Martin Gardner in 1976 to call them snark, a mysterious creature. In 1975, Rufus Isaacs introduced a method known as dot product, which allowed the construction of new snarks from known snarks, and presented the first infinite family of snarks. A new method proposed in 1996 by Martin Kochol allowed to obtaining new snarks from smaller graphs, known as Kochol superposition. However, this method was usually used to obtain snarks with large girth. We applied the Kochol superposition to known snarks: the family of Goldberg snarks (known to be Type 1) and a girth 4 snark recently discovered by Gunnar Brinkmann et al. (known to be Type 2). Surprisingly, when we apply the Kochol superposition to a Type 1 snark with a Type 2 snark, we can obtain new families of snarks of distinct Types: Type 1 and Type 2.

Graph pebbling is a problem in which pebbles are distributed across the vertices of a graph and moved according to a specific rule: two pebbles are removed from a vertex to place one on an adjacent vertex. The goal is to determine the minimum number of pebbles required to ensure that any target vertex can be reached, known as the pebbling number. Computing the pebbling number lies beyond NP in the polynomial hierarchy, leading to bounding methods. One of the most prominent techniques for upper bounds is the Weight Function Lemma (WFL), which relies on costly integer linear optimization. To mitigate this cost, an alternative approach is to consider the dual formulation of the problem, which allows solutions to be constructed by hand through the selection of strategies given by subtrees with associated weight functions. To improve the bounds, the weights should be distributed as uniformly as possible among the vertices, balancing their individual contribution. However, despite its simplicity, this approach lacks a formal framework. To fill this gap, we introduce a novel heuristic method that refines the selection of balanced strategies. The method is motivated by our theoretical analysis of the limitations of the dual approach, in which we prove lower bounds on the best bounds achievable. Our theoretical analysis shows that the bottleneck lies in the farthest vertices from the target, forcing surplus weight onto the closer neighborhoods. To minimize surplus weight beyond the theoretical minimum, our proposed heuristic prioritizes weight assignment to the farthest vertices, building the subtrees starting from the shortest paths to them and then filling in the weights for the remaining vertices. Applying our heuristic to Flower snarks and Blanuša snarks, we improve the best-known upper bounds, demonstrating the effectiveness of a structured strategy selection when using WFL.

Abstract Chordal graphs are the intersection graphs of subtrees of a tree, while interval graphs of subpaths of a path. Undirected path graphs, directed path graphs and rooted directed path graphs are intermediate graph classes, defined, respectively, as the intersection graphs of paths of a tree, of directed paths of an oriented tree, and of directed paths of an out branching. All of these path graphs have vertex leafage 2. Dominating Set, Connected Dominating Set, and Steiner tree problems are ‐hard parameterized by the size of the solution on chordal graphs, ‐complete on undirected path graphs, and polynomial‐time solvable on rooted directed path graphs, and hence also on interval graphs. We further investigate the (parameterized) complexity of all these problems when constrained to chordal graphs, taking the vertex leafage and the aforementioned classes into consideration. We prove that Dominating Set, Connected Dominating Set, and Steiner tree are on chordal graphs when parameterized by the size of the solution plus the vertex leafage, and that Weighted Connected Dominating Set is polynomial‐time solvable on strongly chordal graphs. We also introduce a new subclass of undirected path graphs, which we call in–out rooted directed path graphs, as the intersection graphs of directed paths of an in–out branching. We prove that Dominating Set, Connected Dominating Set, and Steiner tree are solvable in polynomial time on this class, generalizing the polynomiality for rooted directed path graphs proved by Booth and Johnson (SIAM J. Comput. 11 (1982), 191‐199.) and by White et al. (Networks 15 (1985), 109‐124.).

In this work we propose the use of path relinking within a hyper-heuristic framework to solve the Generalised Time-Dependent Asymmetric Traveling Salesman problem applied to Air Travel. We implemented several heuristic selection methods, such as Simple Random, Random Descent, Random Permutation and Reinforcement Learning. We were able to achieve very good solutions for 13 of the 14 instances from a well-known benchmark set, evidencing that the adequate use of a hyper-heuristic framework with path relinking can be very efficient in improving the final results.

A k-total coloring of a graph G is an assignment of k colors to the elements (vertices and edges) of G so that adjacent or incident elements have different colors. The total chromatic number is the smallest integer k for which G has a k-total coloring. The well known Total Coloring Conjecture states that the total chromatic number of a graph is either ∆(G) + 1 or ∆(G) + 2, where ∆(G) is the maximum degree of G. We consider the direct product of complete graphs Km × Kn. It is known that if at least one of the numbers m or n is even, then Km × Kn has total chromatic number equal to ∆(Km × Kn) + 1, except when m = n = 2. We prove that the graph Km × Kn has total chromatic number equal to ∆(Km × Kn) + 1 when both m and n are odd numbers, ensuring in this way that all graphs Km × Kn have total chromatic number equal to ∆ (Km × Kn) + 1, except when m = n = 2.

Abstract Even, Itai, and Shamir (1976) proved simple two‐commodity integral flow is NP‐complete both in the directed and undirected cases. In particular, the directed case was shown to be NP‐complete even if one demand is unitary, which was improved by Fortune, Hopcroft and Wyllie (1980) who proved the problem is still NP‐complete if both demands are unitary. The undirected case, on the other hand, was proved by Robertson and Seymour (1995) to be polynomial‐time solvable if both demands are constant. Nevertheless, the complexity of the undirected case with exactly one constant demand has remained unknown. We close this 40‐year complexity gap, by showing the undirected case is NP‐complete even if exactly one demand is unitary. As a by product, we obtain the NP‐completeness of determining whether a graph contains 1 + d pairwise vertex‐disjoint paths, such that one path is between a given pair of vertices and d paths are between a second given pair of vertices. Additionally, we investigate the complexity of another related network design problem called strict terminal connection.

Sorting by transpositions is a challenging classic problem proposed in genome rearrangement and recently settled as NP-hard. Although the proven hard to sort $3$-permutations are close to the identity, the historical approach has been to study distant permutations, possible candidates to be diametral. The transposition diameter is a related challenging problem, known only for $n \leq 15$. We advance the study of both transposition distance and diameter by considering lonely permutations and the union operation. We present tighter bounds for the distance of lonely $3$-permutations, $u_{n,n-1}$, $u_{n,\frac{n}{2}}$, $u_{n,3}$, and $u_{n,4}$. We set the current lower bound for the transposition diameter back to $\big\lfloor\frac{n+1}{2}\big\rfloor+1$ and propose an alternative union of lonely permutations contributing to the approach used so far in the literature.

Abstract The total chromatic number χT(G) is the least number of colors sufficient to color the elements (vertices and edges) of a graph G in such a way that no incident or adjacent elements receive the same color. In the present work, we obtain two results on total‐coloring. First, we extend the set of partial‐grids classified with respect to the total‐chromatic number, by proving that every 8‐chordal partial‐grid of maximum degree 3 has total chromatic number 4. Second, we prove a result on list‐total‐coloring biconnected outerplanar graphs. If for each element x of a biconnected outerplanar graph G there exists a set Lx of colors such that |Luw| = max{deg(u) + 1, deg(w) + 1} for each edge uw and |Lv| = 7 − δdeg(v),3 − 2δdeg(v),2 (where δi,j = 1 if i = j and δi,j = 0 if i ≠ j) for each vertex v, then there is a total‐coloring π of graph G such that π(x) ∈ Lx for each element x of G. The technique used in these two results is a decomposition by a cutset of two adjacent vertices, whose properties are discussed in the article. © 2011 Wiley Periodicals, Inc. NETWORKS, 2011

H. Eriksson et al. made a breakthrough to the problem of sorting by transpositions by proposing a quotient structure named toric graph, which allowed the reduction of the search space, establishing the transposition diameter $D_t(n)=\lfloor\frac{n+1}{2}\rfloor+1$, for the cases $n=13$ and $n=15$, and invalidating a conjecture by J. Meidanis, M. E. M. T. Walter, and Z. Dias that the transposition diameter would be equal to the transposition distance of the reverse permutation $\lfloor n/2\rfloor+1$. I. Elias and T. Hartman extended the lower bound $D_t(n)\geq\lfloor\frac{n+1}{2}\rfloor+1$, to all odd values of n, $n\geq13$. The value $n=15$ is the largest for which $D_t(n)$ is known. The goal of the present paper is to further study the toric graph, focusing on the case when $n+1$ is prime, providing positive evidence that J. Meidanis, M. E. M. T. Walter, and Z. Dias's conjecture is still valid when n is even. We show that, when $n+1$ is prime, the properties of the reverse permutation are shared by permutations that fall into unitary toric classes; we prove that their reality and desire diagrams have just one cycle, consequently proving that those permutations are separated by at least $n/2$ transpositions among themselves, and we show that there are at least two permutations whose transposition distance is $n/2$ and two permutations, other than the reverse, whose distance is at least $n/2+1$, with respect to the identity.

A bull is a graph with five vertices a,b,c,d,e and five edges ab, ac, bc, da, eb. Here we present polynomial-time combinatorial algorithms for the optimal weighted coloring and weighted clique problems in bull-free perfect graphs. The algorithms are based on a structural analysis and decomposition of bull-free perfect graphs.