The analogue of Hadwiger’s conjecture for the immersion order states that every graph G contains KX(G) as an immersion. Our work is motivated by a strengthening of this conjecture which asserts that every graph G contains KX(G) as a totally odd immersion. As evidence for this strengthened conjecture and inspired by a result of Steiner (2024), we show that if the chromatic number of G is equal to any of its topological lower bounds, then G contains a totally odd immersion of K[x(G)/2]+1.Kneser graphs are canonical examples of graphs satisfying such equalities for their chromatic numbers. Simonyi and Zsban (2010) showed that every Kneser graph G with large enough order (compared to x(G)) contains a totally odd subdivision of KX(G), thus satisfying the motivating conjecture in a strong sense. We show that, in fact, for every t ≥ 8, there are t-chromatic Kneser graphs that contain arbitrarily large complete totally odd subdivisions.
An M9-partition of a graph is a partition of its vertex set into an independent set and a set inducing a complete bipartite graph, i.e. an M9-partition is a 3-coloring (I, J, K) with the additional restriction that each vertex in part J is adjacent to all vertices in part K. A counipolar partition of a graph is a partition of its vertex set into an independent set and a set inducing a complete multipartite graph.In this work, we present partial results on the structure of the families of P5-free graphs admitting either a counipolar or an M9 -partition. Specifically, we focus on forbidden induced subgraph characterizations and recognition algorithms for these graph classes.
In this contribution, we study the total {k}-domination number on graphs. We establish a general upper bound for this number and provide sufficient conditions on a graph to satisfy it at equality. Moreover, for the family of caterpillar graphs, this bound is also tight.The total {k}-domination problem consists of finding a function of minimum value, defined on a set of vertices in a graph, such that in any open neighborhood it has value at least k. We focus on this problem in caterpillar graphs and show that, on this family, it can be reduced to the usual total domination problem (k = 1). Then, we present a representation of a caterpillar in terms of the number of vertices of degree 3 (parents) in it, and the length of the paths induced between two consecutive parents in the central path of the caterpillar. Using this representation, we establish the main result of this work: the value of the total {k}-domination number of every caterpillar, for all k.
A permutation snark is a cubic graph which has a 2-factor consisting of two chordless cycles and is not 3-edge-colourable. The order of every permutation snark is twice an odd number, and thus it is an integer congruent to either 2 or 6 (mod 8). While there exist infinitely many permutation snarks whose orders cover all the integers congruent to 2 (mod 8), no permutation snarks of order 6 (mod 8) are known. Exhaustive computer searches (Brinkmann et al., 2013, and Goedgebeur and Renders, 2025) have revealed that if such a permutation snark exists, then its order must be at least 54. Brinkmann et al. (2013) also asked whether all permutation snarks have order 2 (mod 8). In this paper we relate the existence of permutation snarks of order 6 (mod 8) to the existence of noncritical permutations snarks, which are also unknown. Our main result states that if there exists a noncritical permutation snark, then there is also one on 6 (mod 8) vertices.
