Related papers: Regular graphs of odd degree are antimagic
A graph $G$ is a link-irregular graph if every two distinct vertices of $G$ have non-isomorphic links. The link of a vertex $v$ in $G$ is the subgraph induced by the neighbors of $v$ in $G$. Ali, Chartrand and Zhang [Discussiones…
A $k$-edge-weighting of $G$ is a mapping $\omega:E(G)\longrightarrow \{1,\ldots,k\}$. The edge-weighting of $G$ naturally induces a vertex-colouring $\sigma_{\omega}:V(G)\longrightarrow \mathbb{N}$ given by$\sigma_{\omega}(v)=\sum_{u\in…
Let $G(V, E)$ be a simple connected graph, with $|E| = \epsilon.$ In this paper, we define an edge-set graph $\mathcal G_G$ constructed from the graph $G$ such that any vertex $v_{s,i}$ of $\mathcal G_G$ corresponds to the $i$-th…
Let $\delta$ and $\Delta$ be the minimum and the maximum degree of the vertices of a simple connected graph $G$, respectively. The distinguishing index of a graph $G$, denoted by $D'(G)$, is the least number of labels in an edge labeling of…
Call a colouring of a graph \emph{distinguishing} if the only automorphism of this graph which preserves said colouring is the identity. Let $H$ be an arbitrary graph. We say that a graph $G$ is \emph{$H$-free} if $G$ does not contain an…
Given a finite simple graph $G$, an odd cover of $G$ is a collection of complete bipartite graphs, or bicliques, in which each edge of $G$ appears in an odd number of bicliques and each non-edge of $G$ appears in an even number of…
Since Sedl\'{a}$\breve{\mbox{c}}$ek introduced the notion of magic labeling of a graph in 1963, a variety of magic labelings of a graph have been defined and studied. In this paper, we study consecutive edge magic labelings of a connected…
A graph $G$ is called edge-magic if there exists a bijective function $f:V\left(G\right) \cup E\left(G\right)\rightarrow \left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert +\left\vert E\left( G\right) \right\vert \right\}$ such…
In this paper, we investigate odd graceful graph, odd strongly harmonious graph, bipartite graph and their relationship. We proved following results: (1) if G is odd strongly harmonious graph, then G is odd graceful graph ;(2) if G is…
In this paper, we prove that every graph with average degree at least $s+t+2$ has a vertex partition into two parts, such that one part has average degree at least $s$, and the other part has average degree at least $t$. This solves a…
Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $p$ and size $q$. For $k\ge 1$, a bijection $f: V(G)\cup E(G) \to \{k, k+1, k+2, \ldots, k+p+q-1\}$ such that $f(uv)= |f(u) - f(v)|$ for every edge $uv\in E(G)$ is said…
A $K_t$-expansion consists of $t$ vertex-disjoint trees, every two of which are joined by an edge. We call such an expansion odd if its vertices can be two-colored so that the edges of the trees are bichromatic but the edges between trees…
A graph is odd if all of its vertices have odd degrees. In particular, an odd spanning tree in a connected graph is a spanning tree in which all vertices have odd degrees. In this paper we establish a unified technique to enumerate odd…
Let $G=(V, E)$ be a connected graph. A bijection $f: E\to \{1, \ldots, |E|\}$ is called a local antimagic labeling if for any two adjacent vertices $x$ and $y$, $f^+(x)\neq f^+(y)$, where $f^+(x)=\sum_{e\in E(x)}f(e)$ and $E(x)$ is the set…
This paper has dual aims. First is to develop practical universal coding methods for unlabeled graphs. Second is to use these for graph anomaly detection. The paper develops two coding methods for unlabeled graphs: one based on the degree…
An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. We define the adjacency, incidence and Laplacian matrices of an oriented hypergraph and study each of them. We extend several matrix…
Liu and Ma [J. Combin. Theory Ser. B, 2018] conjectured that every $2$-connected non-bipartite graph with minimum degree at least $k+1$ contains $\lceil k/2\rceil $ cycles with consecutive odd lengths. In particular, they showed that this…
A classical result by Otter shows that the complete graph has an exponential number of non-isomorphic spanning trees. This was recently extended by Lee to every almost regular graph of sufficiently large degree. In this paper, we consider…
An edge-colouring of a graph is distinguishing, if the only automorphism which preserves the colouring is the identity. It has been conjectured that all but finitely many connected, finite, regular graphs admit a distinguishing…
A normal partition of the edges of a cubic graph is a partition into trails (no repeated edge) such that each vertex is the end vertex of exactly one trail of the partition. We investigate this notion and give some results and problems.