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Register a new userGraphs of large linear size are antimagic
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Authors
Tom Eccles
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Given a graph $G=(V,E)$ and a colouring $f:Emapsto mathbb N$, the induced colour of a vertex $v$ is the sum of the colours at the edges incident with $v$. If all the induced colours of vertices of $G$ are distinct, the colouring is called antimagic. If $G$ has a bijective antimagic colouring $f:Emapsto {1,dots,|E|}$, the graph $G$ is called antimagic. A conjecture of Hartsfield and Ringel states that all connected graphs other than $K_2$ are antimagic. Alon, Kaplan, Lev, Roddity and Yuster proved this conjecture for graphs with minimum degree at least $c log |V|$ for some constant $c$; we improve on this result, proving the conjecture for graphs with average degree at least some constant $d_0$.
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Given a digraph $D$ with $m $ arcs, a bijection $tau: A(D)rightarrow {1, 2, ldots, m}$ is an antimagic labeling of $D$ if no two vertices in $D$ have the same vertex-sum, where the vertex-sum of a vertex $u $ in $D$ under $tau$ is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. We say $(D, tau)$ is an antimagic orientation of a graph $G$ if $D$ is an orientation of $G$ and $tau$ is an antimagic labeling of $D$. Motivated by the conjecture of Hartsfield and Ringel from 1990 on antimagic labelings of graphs, Hefetz, M{u}tze, and Schwartz in 2010 initiated the study of antimagic orientations of graphs, and conjectured that every connected graph admits an antimagic orientation. This conjecture seems hard, and few related results are known. However, it has been verified to be true for regular graphs and biregular bipartite graphs. In this paper, we prove that every connected graph $G$ on $nge9$ vertices with maximum degree at least $n-5$ admits an antimagic orientation.
A graph $G$ is $k$-$weighted-list-antimagic$ if for any vertex weighting $omegacolon V(G)tomathbb{R}$ and any list assignment $Lcolon E(G)to2^{mathbb{R}}$ with $|L(e)|geq |E(G)|+k$ there exists an edge labeling $f$ such that $f(e)in L(e)$ for all $ein E(G)$, labels of edges are pairwise distinct, and the sum of the labels on edges incident to a vertex plus the weight of that vertex is distinct from the sum at every other vertex. In this paper we prove that every graph on $n$ vertices having no $K_1$ or $K_2$ component is $lfloor{frac{4n}{3}}rfloor$-weighted-list-antimagic. An oriented graph $G$ is $k$-$oriented-antimagic$ if there exists an injective edge labeling from $E(G)$ into ${1,dotsc,|E(G)|+k}$ such that the sum of the labels on edges incident to and oriented toward a vertex minus the sum of the labels on edges incident to and oriented away from that vertex is distinct from the difference of sums at every other vertex. We prove that every graph on $n$ vertices with no $K_1$ component admits an orientation that is $lfloor{frac{2n}{3}}rfloor$-oriented-antimagic.
A $labeling$ of a digraph $D$ with $m$ arcs is a bijection from the set of arcs of $D$ to ${1,2,ldots,m}$. A labeling of $D$ is $antimagic$ if no two vertices in $D$ have the same vertex-sum, where the vertex-sum of a vertex $u in V(D)$ for a labeling is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. An antimagic orientation $D$ of a graph $G$ is $antimagic$ if $D$ has an antimagic labeling. Hefetz, M$ddot{u}$tze and Schwartz in [J. Graph Theory 64(2010)219-232] raised the question: Does every graph admits an antimagic orientation? It had been proved that for any integer $d$, every 2$d$-regular graph with at most two odd components has an antimagic orientation. In this paper, we consider the 2$d$-regular graph with many odd components. We show that every 2$d$-regular graph with any odd components has an antimagic orientation provide each odd component with enough order.
Given a digraph $D$ with $m$ arcs and a bijection $tau: A(D)rightarrow {1, 2, ldots, m}$, we say $(D, tau)$ is an antimagic orientation of a graph $G$ if $D$ is an orientation of $G$ and no two vertices in $D$ have the same vertex-sum under $tau$, where the vertex-sum of a vertex $u$ in $D$ under $tau$ is the sum of labels of all arcs entering $u$ minus the sum of labels of all arcs leaving $u$. Hefetz, M{u}tze, and Schwartz in 2010 initiated the study of antimagic orientations of graphs, and conjectured that every connected graph admits an antimagic orientation. This conjecture seems hard, and few related results are known. However, it has been verified to be true for regular graphs, biregular bipartite graphs, and graphs with large maximum degree. In this paper, we establish more evidence for the aforementioned conjecture by studying antimagic orientations of graphs $G$ with independence number at least $|V(G)|/2$ or at most four. We obtain several results. The method we develop in this paper may shed some light on attacking the aforementioned conjecture.
Motivated by the conjecture of Hartsfield and Ringel on antimagic labelings of undirected graphs, Hefetz, M{u}tze, and Schwartz initiated the study of antimagic labelings of digraphs in 2010. Very recently, it has been conjectured in [Antimagic orientation of even regular graphs, J. Graph Theory, 90 (2019), 46-53.] that every graph admits an antimagtic orientation, which is a strengthening of an earlier conjecture of Hefetz, M{u}tze and Schwartz. In this paper, we prove that every $2d$-regular graph (not necessarily connected) admits an antimagic orientation, where $dge2$. Together with known results, our main result implies that the above-mentioned conjecture is true for all regular graphs.
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