Cycles with consecutive odd lengths

It is proved that there exists an absolute constant c > 0 such that for every natural number k, every non-bipartite 2-connected graph with average degree at least ck contains k cycles with consecutive odd lengths. This implies the existence of the absolute constant d > 0 that every non-bipartite 2-connected graph with minimum degree at least dk contains cycles of all lengths modulo k, thus providing an answer (in a strong form) to a question of Thomassen. Both results are sharp up to the constant factors.Comment: 7 page

Track Layouts of Graphs

A \emph{$(k,t)$-track layout} of a graph $G$ consists of a (proper) vertex $t$-colouring of $G$, a total order of each vertex colour class, and a (non-proper) edge $k$-colouring such that between each pair of colour classes no two monochromatic edges cross. This structure has recently arisen in the study of three-dimensional graph drawings. This paper presents the beginnings of a theory of track layouts. First we determine the maximum number of edges in a $(k,t)$-track layout, and show how to colour the edges given fixed linear orderings of the vertex colour classes. We then describe methods for the manipulation of track layouts. For example, we show how to decrease the number of edge colours in a track layout at the expense of increasing the number of tracks, and vice versa. We then study the relationship between track layouts and other models of graph layout, namely stack and queue layouts, and geometric thickness. One of our principle results is that the queue-number and track-number of a graph are tied, in the sense that one is bounded by a function of the other. As corollaries we prove that acyclic chromatic number is bounded by both queue-number and stack-number. Finally we consider track layouts of planar graphs. While it is an open problem whether planar graphs have bounded track-number, we prove bounds on the track-number of outerplanar graphs, and give the best known lower bound on the track-number of planar graphs.Comment: The paper is submitted for publication. Preliminary draft appeared as Technical Report TR-2003-07, School of Computer Science, Carleton University, Ottawa, Canad

Full Orientability of the Square of a Cycle

Let D be an acyclic orientation of a simple graph G. An arc of D is called dependent if its reversal creates a directed cycle. Let d(D) denote the number of dependent arcs in D. Define m and M to be the minimum and the maximum number of d(D) over all acyclic orientations D of G. We call G fully orientable if G has an acyclic orientation with exactly k dependent arcs for every k satisfying m <= k <= M. In this paper, we prove that the square of a cycle C_n of length n is fully orientable except n=6.Comment: 7 pages, accepted by Ars Combinatoria on May 26, 201

Steiner trees for hereditary graph classes.

We consider the classical problems (Edge) Steiner Tree and Vertex Steiner Tree after restricting the input to some class of graphs characterized by a small set of forbidden induced subgraphs. We show a dichotomy for the former problem restricted to (H1,H2) -free graphs and a dichotomy for the latter problem restricted to H-free graphs. We find that there exists an infinite family of graphs H such that Vertex Steiner Tree is polynomial-time solvable for H-free graphs, whereas there exist only two graphs H for which this holds for Edge Steiner Tree. We also find that Edge Steiner Tree is polynomial-time solvable for (H1,H2) -free graphs if and only if the treewidth of the class of (H1,H2) -free graphs is bounded (subject to P≠NP ). To obtain the latter result, we determine all pairs (H1,H2) for which the class of (H1,H2) -free graphs has bounded treewidth

A solution to Erd\H{o}s and Hajnal's odd cycle problem

In 1981, Erd\H{o}s and Hajnal asked whether the sum of the reciprocals of the odd cycle lengths in a graph with infinite chromatic number is necessarily infinite. Let $\mathcal{C}(G)$ be the set of cycle lengths in a graph $G$ and let $\mathcal{C}_\text{odd}(G)$ be the set of odd numbers in $\mathcal{C}(G)$. We prove that, if $G$ has chromatic number $k$, then $\sum_{\ell\in \mathcal{C}_\text{odd}(G)}1/\ell\geq (1/2-o_k(1))\log k$. This solves Erd\H{o}s and Hajnal's odd cycle problem, and, furthermore, this bound is asymptotically optimal. In 1984, Erd\H{o}s asked whether there is some $d$ such that each graph with chromatic number at least $d$ (or perhaps even only average degree at least $d$) has a cycle whose length is a power of 2. We show that an average degree condition is sufficient for this problem, solving it with methods that apply to a wide range of sequences in addition to the powers of 2. Finally, we use our methods to show that, for every $k$, there is some $d$ so that every graph with average degree at least $d$ has a subdivision of the complete graph $K_k$ in which each edge is subdivided the same number of times. This confirms a conjecture of Thomassen from 1984.Comment: 42 pages, 3 figures. Version accepted for publicatio

Balanced-chromatic number and Hadwiger-like conjectures

Motivated by different characterizations of planar graphs and the 4-Color Theorem, several structural results concerning graphs of high chromatic number have been obtained. Toward strengthening some of these results, we consider the \emph{balanced chromatic number}, $\chi_b(\hat{G})$, of a signed graph $\hat{G}$. This is the minimum number of parts into which the vertices of a signed graph can be partitioned so that none of the parts induces a negative cycle. This extends the notion of the chromatic number of a graph since $\chi(G)=\chi_b(\tilde{G})$, where $\tilde{G}$ denotes the signed graph obtained from~$G$ by replacing each edge with a pair of (parallel) positive and negative edges. We introduce a signed version of Hadwiger's conjecture as follows. Conjecture: If a signed graph $\hat{G}$ has no negative loop and no $\tilde{K_t}$-minor, then its balanced chromatic number is at most $t-1$. We prove that this conjecture is, in fact, equivalent to Hadwiger's conjecture and show its relation to the Odd Hadwiger Conjecture. Motivated by these results, we also consider the relation between subdivisions and balanced chromatic number. We prove that if $(G, \sigma)$ has no negative loop and no $\tilde{K_t}$-subdivision, then it admits a balanced $\frac{79}{2}t^2$-coloring. This qualitatively generalizes a result of Kawarabayashi (2013) on totally odd subdivisions