A mapping from the vertex set of a graph G = (V,E) into an interval of integers {0,...,k} is an L(2,1)-labelling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a common neighbour are mapped onto distinct integers. It is known that for any fixed k >= 4, deciding the existence of such a labelling is an NP-complete problem while it is polynomial for k <= 3. For even k >= 8, it remains NP-complete when restricted to planar graphs. In this paper, we show that it remains NP-complete for any k >= 4 by reduction from Planar Cubic Two-Colourable Perfect Matching. Schaefer stated without proof that Planar Cubic Two-Colourable Perfect Matching is NP-complete. In this paper we give a proof of this.
We show that there exists an adjacency labelling scheme for planar graphs where each vertex of an $n$-vertex planar graph $G$ is assigned a $(1+o(1))\log_2 n$-bit label and the labels of two vertices $u$ and $v$ are sufficient to determine if $uv$ is an edge of $G$. This is optimal up to the lower order term and is the first such asymptotically optimal result. An alternative, but equivalent, interpretation of this result is that, for every $n$, there exists a graph $U_n$ with $n^{1+o(1)}$ vertices such that every $n$-vertex planar graph is an induced subgraph of $U_n$. These results generalize…
The ($d$,1)-total labelling of graphs was introduced by Havet and Yu. In this paper, we prove that, for planar graph $G$ with maximum degree $Δ\geq12$ and $d=2$, the (2,1)-total labelling number $λ_2^T(G)$ is at most $Δ+2$.
A magic labelling of a graph $G$ with magic sum $s$ is a labelling of the edges of $G$ by nonnegative integers such that for each vertex $v\in V$, the sum of labels of all edges incident to $v$ is equal to the same number $s$. Stanley gave remarkable results on magic labellings, but the distinct labelling case is much more complicated. We consider the complete construction of all magic labellings of a given graph $G$. The idea is illustrated in detail by dealing with three regular graphs. We give combinatorial proofs. The structure result was used to enumerate the corresponding magic distinct …
A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. In this paper, we first give a useful structural theorem for 1-planar graphs, and then apply it to the list edge and list total coloring, the $(p,1)$-total labelling, and the equitable edge coloring of 1-planar graphs. More precisely, we verify the well-known List Edge Coloring Conjecture and List Total Coloring Conjecture for 1-planar graph with maximum degree at least 18, prove that the $(p,1)$-total labelling number of every 1-planar graph $G$ is at most $Δ(G)+2p-2$ provided that $Δ(G)\…
A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares. Since then rainbow structures have been the focus of extensive research and have found applications in the areas of graph labelling and decomposition. An edge-colouring is locally $k$-bounded if each vertex is contained in at most $k$ edges of the same colour. In this paper we prove that any such edge-colouring of the complete graph $K_n$ contains a rainbow copy of every tree with at most $(1-o…
The planar graph product structure theorem of Dujmović, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] states that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. This result has been the key tool to resolve important open problems regarding queue layouts, nonrepetitive colourings, centered colourings, and adjacency labelling schemes. In this paper, we extend this line of research by utilizing shallow minors to prove analogous product structure theorems for several beyond planar graph classes. The key observation that drives our work …
Adjacency labelling for proper minor-closed graph classes
University of Ottawa（渥太华大学）；LaBRI, University of Bordeaux（波尔多大学 LaBRI）；Université libre de Bruxelles（布鲁塞尔自由大学）；Jagiellonian University（雅盖隆大学）；Carleton University（卡尔顿大学）；Monash University（蒙纳士大学）
A proper labelling of a graph $G$ is a pair $(π,c_π)$ in which $π$ is an assignment of numeric labels to some elements of $G$, and $c_π$ is a colouring induced by $π$ through some mathematical function over the set of labelled elements. In this work, we consider gap-vertex-labellings, in which the colour of a vertex is determined by a function considering the largest difference between the labels assigned to its neighbours. We present the first upper-bound for the vertex-gap number of arbitrary graphs, which is the least number of labels required to properly label a graph. We investigate famil…
Given a finite or infinite graph $G$ and positive integers $\ell, h_1, h_2, h_3$, an $L(h_1, h_2, h_3)$-labelling of $G$ with span $\ell$ is a mapping $f: V(G) \rightarrow \{0, 1, 2, \ldots, \ell\}$ such that, for $i = 1, 2, 3$ and any $u, v \in V(G)$ at distance $i$ in $G$, $|f(u) - f(v)| \geq h_i$. A $C(h_1, h_2, h_3)$-labelling of $G$ with span $\ell$ is defined similarly by requiring $|f(u) - f(v)|_{\ell} \ge h_i$ instead, where $|x|_{\ell} = \min\{|x|, \ell-|x|\}$. The minimum span of an $L(h_1, h_2, h_3)$-labelling, or a $C(h_1, h_2, h_3)$-labelling, of $G$ is denoted by $λ_{h_1,h_2,h_3}…
A distinguishing r-vertex-labelling (resp. r-edge-labelling) of an undirected graph G is a mapping $λ$ from the set of vertices (resp. the set of edges) of G to the set of labels {1,. .. , r} such that no non-trivial automorphism of G preserves all the vertex (resp. edge) labels. The distinguishing number D(G) and the distinguishing index D (G) of G are then the smallest r for which G admits a distinguishing r-vertex-labelling or r-edge-labelling, respectively. The distinguishing chromatic number D $χ$ (G) and the distinguishing chromatic index D $χ$ (G) are defined similarly, with the additio…
Dujmović et al (FOCS2019) recently proved that every planar graph $G$ is a subgraph of $H\boxtimes P$, where $\boxtimes$ denotes the strong graph product, $H$ is a graph of treewidth 8 and $P$ is a path. This result has found numerous applications to linear graph layouts, graph colouring, and graph labelling. The proof given by Dujmović et al is based on a similar decomposition of Pilipczuk and Siebertz (SODA2019) which is constructive and leads to an $O(n^2)$ time algorithm for finding $H$ and the mapping from $V(G)$ onto $V(H\boxtimes P)$. In this note, we show that this algorithm can be mad…
This paper establishes connections between the structure of a semigroup and the minimum spans of distance labellings of its Cayley graphs. We show that certain general restrictions on the minimum spans are equivalent to the semigroup being combinatorial, and that other restrictions are equivalent to the semigroup being a right zero band. We obtain a description of the structure of all semigroups $S$ and their subsets $C$ such that $\Cay(S,C)$ is a disjoint union of complete graphs, and show that this description is also equivalent to several restrictions on the minimum span of $\Cay(S,C)$. We …
The concept of sum labelling was introduced in 1990 by Harary. A graph is a sum graph if its vertices can be labelled by distinct positive integers in such a way that two vertices are connected by an edge if and only if the sum of their labels is the label of another vertex in the graph. It is easy to see that every sum graph has at least one isolated vertex, and every graph can be made a sum graph by adding at most $n^2$ isolated vertices to it. The minimum number of isolated vertices that need to be added to a graph to make it a sum graph is called the sum number of the graph. The sum number…
In this paper, we study the achromatic and the pseudoachromatic numbers of planar and outerplanar graphs as well as planar graphs of girth 4 and graphs embedded on a surface. We give asymptotically tight results and lower bounds for maximal embedded graphs.
The Graceful Tree Conjecture of Rosa from 1967 asserts that the vertices of each tree T of order n can be injectively labelled by using the numbers {1,2,...,n} in such a way that the absolute differences induced on the edges are pairwise distinct. We prove the following relaxation of the conjecture for each c>0 and for all n>n_0(c). Suppose that (i) the maximum degree of T is bounded by O(n/log n), and (ii) the vertex labels are chosen from the set {1,2,..., (1+c)n}. Then there is an injective labelling of V(T) such that the absolute differences on the edges are pairwise distinct. In particula…
Let $G=(V,E)$ be a graph without isolated vertices. A set $S\subseteq V$ is a paired-domination set if every vertex in $V-S$ is adjacent to a vertex in $S$ and the subgraph induced by $S$ contains a perfect matching. The paired-domination problem is to determine the paired-domination number, which is the minimum cardinality of a paired-dominating set. Motivated by a mistaken algorithm given by Chen, Kang and Ng [ Paired domination on interval and circular-arc graphs, Disc. Appl. Math. 155(2007),2077-2086], we present two linear time algorithms to find a minimum cardinality paired-dominating se…
For all $k \geq 1$, we show that deciding whether a graph is $k$-planar is NP-complete, extending the well-known fact that deciding 1-planarity is NP-complete. Furthermore, we show that the gap version of this decision problem is NP-complete. In particular, given a graph with local crossing number either at most $k\ge 1$ or at least $2k$, we show that it is NP-complete to decide whether the local crossing number is at most $k$ or at least $2k$. This algorithmic lower bound proves the non-existence of a $(2-ε)$-approximation algorithm for any fixed $k \ge 1$. In addition, we analyze the sometim…
The $γ$-graph of a graph $G$ is the graph whose vertices are labelled by the minimum dominating sets of $G$, in which two vertices are adjacent when their corresponding minimum dominating sets (each of size $γ(G)$) intersect in a set of size $γ(G)-1$. We extend the notion of a $γ$-graph from distance-1-domination to distance-$d$-domination, and ask which graphs $H$ occur as $γ$-graphs for a given value of~$d \ge 1$. We show that, for all $d$, the answer depends only on whether the vertices of $H$ admit a labelling consistent with the adjacency condition for a conventional $γ$-graph. This resul…
It is a well-known fact that hamiltonicity in planar cubic graphs is an NP-complete problem. This implies that the existence of an A-trail in plane eulerian graphs is also an NP-complete problem even if restricted to planar 3-connected eulerian graphs. In this paper we deal with hamiltonicity in planar cubic graphs G having a facial 2-factor Q via (quasi) spanning trees of faces in G/Q and study the algorithmic complexity of finding such (quasi) spanning trees of faces. We show, in particular, that if Barnette's Conjecture is false, then hamiltonicity in 3-connected planar cubic bipartite grap…
Huynh et al. recently showed that a countable graph $G$ which contains every countable planar graph as a subgraph must contain arbitrarily large finite complete graphs as topological minors, and an infinite complete graph as a minor. We strengthen this result by showing that the same conclusion holds, if $G$ contains every countable planar graph as a topological minor. In particular, there is no countable planar graph containing every countable planar graph as a topological minor, answering a question by Diestel and Kühn. Moreover, we construct a locally finite planar graph which contains ever…
We examine the effect of bounding the diameter for well-studied variants of the Colouring problem. A colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring. The last problem is also known as $L(1,1)$-Labelling and we also consider the framework of $L(a,b)$-Labelling. We prove a number of (almost-)complete complexity classifications. In particular, we show that for graphs of diameter at most $d$, Acyclic…
A graph G is (d_1,..,d_l)-colorable if the vertex set of G can be partitioned into subsets V_1,..,V_l such that the graph G[V_i] induced by the vertices of V_i has maximum degree at most d_i for all 1 <= i <= l. In this paper, we focus on complexity aspects of such colorings when l=2,3. More precisely, we prove that, for any fixed integers k,j,g with (k,j) distinct form (0,0) and g >= 3, either every planar graph with girth at least g is (k,j)-colorable or it is NP-complete to determine whether a planar graph with girth at least g is (k,j)-colorable. Also, for any fixed integer k, it is NP-com…
Given a connected, undirected graph whose edges are labelled (or coloured), the minimum labelling spanning tree (MLST) problem seeks a spanning tree whose edges have the smallest number of distinct labels (or colours). In recent work, the MLST problem has been shown to be NP-hard and some effective heuristics have been proposed and analyzed. In this paper we present an intelligent optimization algorithm to solve the problem. It is obtained by the basic Variable Neighbourhood Search heuristic with the integration of other complements from machine learning, statistics and experimental algorithmi…
In 1896 Heinrich Maschke characterized planar finite groups, that is groups which admit a generating system such that the resulting Cayley graph is planar. In our study we consider the question, which finite semigroups have a planar Cayley graph. Right groups are a class of semigroups relatively close to groups. We present a complete characterization of planar right groups.
A proof labelling scheme for a graph class $\mathcal{C}$ is an assignment of certificates to the vertices of any graph in the class $\mathcal{C}$, such that upon reading its certificate and the certificates of its neighbors, every vertex from a graph $G\in \mathcal{C}$ accepts the instance, while if $G\not\in \mathcal{C}$, for every possible assignment of certificates, at least one vertex rejects the instance. It was proved recently that for any fixed surface $Σ$, the class of graphs embeddable in $Σ$ has a proof labelling scheme in which each vertex of an $n$-vertex graph receives a certifica…
Motivated by applications in graph drawing and information visualization, we examine the planar split thickness of a graph, that is, the smallest $k$ such that the graph is $k$-splittable into a planar graph. A $k$-split operation substitutes a vertex $v$ by at most $k$ new vertices such that each neighbor of $v$ is connected to at least one of the new vertices. We first examine the planar split thickness of complete graphs, complete bipartite graphs, multipartite graphs, bounded degree graphs, and genus-1 graphs. We then prove that it is NP-hard to recognize graphs that are $2$-splittable int…
For a planar graph with a given f-vector $(f_{0}, f_{1}, f_{2}),$ we introduce a cubic polynomial whose coefficients depend on the f-vector. The planar graph is said to be real if all the roots of the corresponding polynomial are real. Thus we have a bipartition of all planar graphs into two disjoint class of graphs, real and complex ones. As a contribution toward a full recognition of planar graphs in this bipartition, we study and recognize completely a subclass of planar graphs that includes all the connected grid subgraphs. Finally, all the 2-connected triangle-free complex planar graphs o…
We present the first combinatorial scheme for counting labelled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function of labelled 4-regular planar graphs can be computed effectively as the solution of a system of equations, from which the coefficients can be extracted. As a byproduct, we also enumerate labelled 3-connected 4-regular planar graphs, and simple 4-regular rooted maps.
Graph packing and partitioning problems have been studied in many contexts, including from the algorithmic complexity perspective. Consider the packing problem of determining whether a graph contains a spanning tree and a cycle that do not share edges. Bernáth and Király proved that this decision problem is NP-complete and asked if the same result holds when restricting to planar graphs. Similarly, they showed that the packing problem with a spanning tree and a path between two distinguished vertices is NP-complete. They also established the NP-completeness of the partitioning problem of deter…
A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by no more than one other edge (and any pair of crossing edges cross only once). A non-1-planar graph $G$ is minimal if the graph $G-e$ is 1-planar for every edge $e$ of $G$. We construct two infinite families of minimal non-1-planar graphs and show that for every integer $n > 62$, there are at least $2^{(n-54)/4}$ nonisomorphic minimal non-1-planar graphs of order $n$. It is also proved that testing 1-planarity is NP-complete.
Given a non-trivial finite Abelian group $(A,+)$, let $n(A) \ge 2$ be the smallest integer such that for every labelling of the arcs of the bidirected complete graph of order $n(A)$ with elements from $A$ there exists a directed cycle for which the sum of the arc-labels is zero. The problem of determining $n(\mathbb{Z}_q)$ for integers $q \ge 2$ was recently considered by Alon and Krivelevich, who proved that $n(\mathbb{Z}_q)=O(q \log q)$. Here we improve their bound and show that $n(\mathbb{Z}_q)$ grows linearly. More generally we prove that for every finite Abelian group $A$ we have $n(A) \l…
A balanced graph is a bipartite graph with no induced circuit of length 2 mod 4. These graphs arise in linear programming. We focus on graph-algebraic properties of balanced graphs to prove a complete classification of balanced Cayley graphs on abelian groups. Moreover, in Section 5 of this paper, we prove that there is no cubic balanced planar graph. Finally, some remarkable conjectures for balanced regular graphs are also presented.
The prism over a graph $G$ is the Cartesian product of $G$ with the complete graph $K_2$. A graph $G$ is hamiltonian if there exists a spanning cycle in $G$, and $G$ is prism-hamiltonian if the prism over $G$ is hamiltonian. In [M.~Rosenfeld, D.~Barnette, Hamiltonian circuits in certain prisms, Discrete Math. 5 (1973), 389--394] the authors conjectured that every 3-connected planar graph is prism-hamiltonian. We construct a counterexample to the conjecture.
The problem Cover(H) asks whether an input graph G covers a fixed graph H (i.e., whether there exists a homomorphism G to H which locally preserves the structure of the graphs). Complexity of this problem has been intensively studied. In this paper, we consider the problem PlanarCover(H) which restricts the input graph G to be planar. PlanarCover(H) is polynomially solvable if Cover(H) belongs to P, and it is even trivially solvable if H has no planar cover. Thus the interesting cases are when H admits a planar cover, but Cover(H) is NP-complete. This also relates the problem to the long-stand…
Recently, the second and third author showed that complete geometric graphs on $2n$ vertices in general cannot be partitioned into $n$ plane spanning trees. Building up on this work, in this paper, we initiate the study of partitioning into beyond planar subgraphs, namely into $k$-planar and $k$-quasi-planar subgraphs and obtain first bounds on the number of subgraphs required in this setting.
The biplanar crossing number of a graph $G$ is the minimum number of crossings over all possible drawings of the edges of $G$ in two disjoint planes. We present new bounds on the biplanar crossing number of complete graphs and complete bipartite graphs. In particular, we prove that the biplanar crossing number of complete bipartite graphs can be approximated to within a factor of $3$, improving over the best previously known approximation factor of $4.03$. For complete graphs, we provide a new approximation factor of $3.17$, improving over the best previous factor of $4.34$. We provide similar…
Here we prove that counting maximum matchings in planar, bipartite graphs is #P-complete. This is somewhat surprising in the light that the number of perfect matchings in planar graphs can be computed in polynomial time. We also prove that counting non-necessarily perfect matchings in planar graphs is already #P-complete if the problem is restricted to bipartite graphs. So far hardness was proved only for general, non-necessarily bipartite graphs.
We classify the planar cubic Cayley graphs of connectivity 2, providing an explicit presentation and embedding for each of them. Combined with [9] this yields a complete description of all planar cubic Cayley graphs.
In this article we consider hook removal operators on odd partitions, i.e., partitions labelling odd-degree irreducible characters of finite symmetric groups. In particular we complete the discussion, started by Isaacs, Navarro, Olsson and Tiep in 2016, concerning the commutativity of such operators.
Dujmović et al. [\emph{J.~ACM}~'20] recently proved that every planar graph is isomorphic to a subgraph of the strong product of a bounded treewidth graph and a path. Analogous results were obtained for graphs of bounded Euler genus or apex-minor-free graphs. These tools have been used to solve longstanding problems on queue layouts, non-repetitive colouring, $p$-centered colouring, and adjacency labelling. This paper proves analogous product structure theorems for various non-minor-closed classes. One noteable example is $k$-planar graphs (those with a drawing in the plane in which each edge …
Beyond-planarity focuses on combinatorial properties of classes of non-planar graphs that allow for representations satisfying certain local geometric or topological constraints on their edge crossings. Beside the study of a specific graph class for its maximum edge density, another parameter that is often considered in the literature is the size of the largest complete or complete bipartite graph belonging to it. Overcoming the limitations of standard combinatorial arguments, we present a technique to systematically generate all non-isomorphic topological representations of complete and compl…
A graph is outer-1-planar if it can be drawn in the plane so that all vertices are on the outer face and each edge is crossed at most once. In this paper, we completely determine the edge chromatic number of outer 1-planar graphs.
A radio labeling of a graph $G$ is a mapping $f$ : $V(G) \rightarrow \{0, 1, 2,...\}$ such that $|f(u)-f(v)| \geq diam(G) + 1 - d(u,v)$ holds for every pair of vertices $u$ and $v$, where $diam(G)$ is the diameter of $G$ and $d(u,v)$ is the distance between $u$ and $v$ in $G$. The radio number of $G$, denoted by $rn(G)$, is the smallest $t$ such that $G$ admits a radio labeling with $t=\max\{|f(v)-f(u)|: v, u \in V(G)\}$. A block graph is a graph such that each block (induced maximal 2-connected subgraph) is a complete graph. In this paper, a lower bound for the radio number of block graphs is…
The product version of the 1-2-3 Conjecture, introduced by Skowronek-Kazi{ó}w in 2012, states that, a few obvious exceptions apart, all graphs can be 3-edge-labelled so that no two adjacent vertices get incident to the same product of labels. To date, this conjecture was mainly verified for complete graphs and 3-colourable graphs. As a strong support to the conjecture, it was also proved that all graphs admit such 4-labellings. In this work, we investigate how a recent proof of the multiset version of the 1-2-3 Conjecture by Vu{\v c}kovi{ć} can be adapted to prove results on the product versio…
The subject of this paper is the study of small complete arcs in $\mathrm{PG}(2,q)$, for $q$ odd, with at least $(q+1)/2$ points on a conic. We give a short comprehensive proof of the completeness problem left open by Segre in his seminal work [20]. This gives an alternative to Pellegrino's long proof which was obtained in a series of papers in the 1980s. As a corollary of our analysis, we obtain a counterexample to a misconception in the literature [6].
We obtain a complete description of the planar cubic Cayley graphs, providing an explicit presentation and embedding for each of them. This turns out to be a rich class, comprising several infinite families. We obtain counterexamples to conjectures of Mohar, Bonnington and Watkins. Our analysis makes the involved graphs accessible to computation, corroborating a conjecture of Droms.
An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. It was proved that computing $rc(G)$ is an NP-Hard problem, as well as that even deciding whether a graph has $rc(G)=2$ is NP-Complete. It is known that deciding whether a given edge-colored graph is rainbow connected is NP-Complete. We will prove that it is still NP-Complete even when the edge-colored graph …
