Related papers: A Simple Proof for the Cycle Double Cover Conjectu…
The disproved Nash Williams conjecture states that every 4-regular 4-connected graph has a hamiltonian cycle. We show that a modification of this conjecture is equivalent to the Dominating Cycle Conjecture.
For a simple graph $G$, denote by $n$, $\Delta(G)$, and $\chi'(G)$ its order, maximum degree, and chromatic index, respectively. A connected class 2 graph $G$ is edge-chromatic critical if $\chi'(G-e)<\Delta(G)+1$ for every edge $e$ of $G$.…
Let ${\cal H}$ denote the family of all graphs with multi-$4$-cycles and suppose that $G \in {\cal H}$. Then, $G$ is a bipartite graph with a vertex bipartition $\{V_{\alpha}, V_{\beta}\}$. We prove that for every vertex $v \in V_{\beta}$…
We show that every edge in a 2-edge-connected planar cubic graph is either contained in a 2-edge-cut or is a chord of some cycle that is contained in a 2-factor of the graph. As a consequence, we show that every edge in a cyclically…
The analogue of Hadwiger's conjecture for the immersion relation states that every graph $G$ contains an immersion of $K_{\chi(G)}$. For graphs with independence number 2, this is equivalent to stating that every such $n$-vertex graph…
The edge clique cover number $ecc(G)$ of a graph $G$ is the size of the smallest set of complete subgraphs whose union covers all edges of $G$. It has been conjectured that all the simple graphs with independence number two satisfy…
The subdivided double construction on 4-regular graphs was used by Poto\v{c}nik and Wilson to explore semi-symmetric (edge-transitive but not vertex-transitive) graphs, and can be used to construct every semi-symmetric 4-regular graph that…
Thomassen, in 1983, conjectured that for a positive integer $k$, every $2$-connected non-bipartite graph of minimum degree at least $k + 1$ contains cycles of all lengths modulo $k$. In this paper, we settle this conjecture affirmatively.
Scott proved in 1997 that for any tree $T$, every graph with bounded clique number which does not contain any subdivision of $T$ as an induced subgraph has bounded chromatic number. Scott also conjectured that the same should hold if $T$ is…
We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lov\'{a}sz from 1969 and Thomassen from 1978, respectively, states that all connected vertex-transitive graphs…
Given a directed graph $D$ of order $n\geq 4$ and a nonempty subset $Y$ of vertices of $D$ such that in $D$ every vertex of $Y$ reachable from every other vertex of $Y$. Assume that for every triple $x,y,z\in Y$ such that $x$ and $y$ are…
A bipartite graph is called bipancyclic if it contains cycles of every even length from four up to the number of vertices in the graph. A theorem of Schmeichel and Mitchem states that for $n \geq 4$, every balanced bipartite graph on $2n$…
A permutation graph is a cubic graph admitting a 1-factor M whose complement consists of two chordless cycles. Extending results of Ellingham and of Goldwasser and Zhang, we prove that if e is an edge of M such that every 4-cycle containing…
In this paper we study an equivalence relation defined on the set of cycle-free $d$-partitions of the complete graph $K_{2d}$. We discuss a conjecture which states that this equivalence relation has only one equivalence class, and show that…
For a graph $G$, let $cp(G)$ denote the minimum number of cliques of $G$ needed to cover the edges of $G$ exactly once. Similarly, let $bp_k(G)$ denote the minimum number of bicliques (i.e. complete bipartite subgraphs of $G$) needed to…
A packing of a graph G with Hamilton cycles is a set of edge-disjoint Hamilton cycles in G. Such packings have been studied intensively and recent results imply that a largest packing of Hamilton cycles in G_n,p a.a.s. has size \lfloor…
Consider the graph that has as vertices all bitstrings of length $2n+1$ with exactly $n$ or $n+1$ entries equal to 1, and an edge between any two bitstrings that differ in exactly one bit. The well-known middle levels conjecture asserts…
We study the class of simple graphs $\mathcal{G}^*$ for which every pair of distinct odd cycles intersect in at most one edge. We give a structural characterization of the graphs in $\mathcal{G}^*$ and prove that every $G \in \mathcal{G}^*$…
In this paper we prove that every sufficiently large 4-edge-connected graph contains the double cycle, $C_{2,r}$, as an immersion. In proving this, we develop a new tool we call a ring-decomposition. We also prove that linear…
A conjecture of Verstra\"ete states that for any fixed $\ell < k$ there exists a positive constant $c$ such that any $C_{2k}$-free graph $G$ contains a $C_{2\ell}$-free subgraph with at least $c |E(G)|$ edges. For $\ell = 2$, this…