Related papers: On The Cycle Double Cover Problem

Hadwiger's Conjecture states that every graph with chromatic number $k$ contains a complete graph on $k$ vertices as a minor. This conjecture is a tremendous strengthening of the Four-Colour Theorem and is regarded as one of the most…

Let $G$ be a graph on $n$ vertices and $C'=v_0v_1\cdots v_{p-1}v_0$ a vertex sequence of $G$ with $p\geq 3$ ($v_i\neq v_j$ for all $i,j=0,1,\ldots,p-1$, $i\neq j$). If for any successive vertices $v_i$, $v_{i+1}$ on $C'$, either…

In a closed 2-cell embedding of a graph each face is homeomorphic to an open disk and is bounded by a cycle in the graph. The Orientable Strong Embedding Conjecture says that every 2-connected graph has a closed 2-cell embedding in some…

We give several sufficient conditions for a double of a free group along a cyclic subgroup to contain a surface subgroup.

For a fixed graph $F,$ the minimum number of edges in an edge-maximal $F$-free subgraph of $G$ is called the $F$-saturation number. The asymptotics of the $F$-saturation number of the binomial random graph $G(n,p)$ for constant $p\in(0,1)$…

In this paper we give a proof of Enomoto's conjecture for graphs of sufficiently large order. Enomoto's conjecture states that, if $G$ is a graph of order $n$ with minimum degree $\delta(G)\geq \frac{n}{2}+1$, then for any pair of vertices…

A classical result of Nosal asserts that every $m$-edge graph with spectral radius $\lambda (G)> \sqrt{m}$ contains a triangle. A celebrated extension of Nikiforov [35] states that if $G$ is an $m$-edge graph with $\lambda (G)> \sqrt{(1-…

A well-known result due to Chvat\'al and Erd\H{o}s (1972) asserts that, if a graph $G$ satisfies $\kappa(G) \ge \alpha(G)$, where $\kappa(G)$ is the vertex-connectivity of $G$, then $G$ has a Hamilton cycle. We prove a similar result…

A graph $G$ is called normal if there exist two coverings, $\mathbb{C}$ and $\mathbb{S}$ of its vertex set such that every member of $\mathbb{C}$ induces a clique in $G$, every member of $\mathbb{S}$ induces an independent set in $G$ and $C…

Reed Conjecture is open for more than 20 years now. Here we prove that Reed Conjecture is valid for (1) {P4UnionK1, Kite}-free graphs (2) {Chair, Kite}-free graphs (3) {K2UnionK2complement , H}-free graphs and (4) {2K2, M}-free graphs where…

We study Hamiltonian circles in the doubly signed complete graph $\Sigma_n = (K_n, \sigma, \mathbb{F}_2^2)$. A circle's double sign is defined as the sum of its edge labels. I establish conditions under which Hamiltonian circles realize all…

For $E \subset {\Bbb F}_q^d$, $d \ge 2$, where ${\Bbb F}_q$ is the finite field with $q$ elements, we consider the distance graph ${\mathcal G}^{dist}_t(E)$, $t \not=0$, where the vertices are the elements of $E$, and two vertices $x$, $y$…

Let $X$ be a general complete intersection of a given multi-degree in a complex projective space. Suppose that the anti-canonical line bundle of $X$ is ample. Using the cylinder homomorphism associated with the family of complete…

We prove for every graph H there exists a>0 such that, for every graph G with at least two vertices, if no induced subgraph of G is a subdivision of H, then either some vertex of G has at least a|G| neighbours, or there are two disjoint…

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…

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…

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…

A path (cycle) in a $2$-edge-colored multigraph is alternating if no two consecutive edges have the same color. The problem of determining the existence of alternating Hamiltonian paths and cycles in $2$-edge-colored multigraphs is an…

A total dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex of $G$ has a neighbor in $D$. A locating-total dominating set of $G$ is a total dominating set $D$ of $G$ with the additional property that every…

In this paper, the concept of cyclic subsets in graph theory is introduced. An interesting theorem which relates to the collective Hamiltonicity of these cyclic subsets in graphs is also presented. This paper uses this theorem to construct…