The chromatic number of an planar graph is not greater than four and this is known by the famous four color theorem and is equal to two when the planar graph is bipartite. When the planar graph is even-triangulated or all cycles are greater than three we know by the Heawood and the Grotszch theorems that the chromatic number is three. There are many conjectures and partial results on three colorability of planar graphs when the graph has specific cycles lengths or cycles with three edges (triangles) have special distance distributions. In this paper we have given a new three colorability criteria for planar graphs that can be considered as an generalization of the Heawood and the Grotszch theorems with respect to the triangulation and cycles of length greater than 3. We have shown that an triangulated planar graph with disjoint holes is 3-colorable if and only if every hole satisfies the parity symmetric property, where a hole is a cycle (face boundary) of length greater than 3.
DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced recently by Dvořák and Postle (2017). In this paper, we prove that every planar graph $G$ without $4$-cycles adjacent to $k$-cycles is DP-$4$-colorable for $k=5$ and $6$. As a consequence, we obtain two new classes of $4$-choosable planar graphs. We use identification of verticec in the proof, and actually prove stronger statements that every pre-coloring of some short cycles can be extended to the whole graph.
It is known that DP-coloring is a generalization of a list coloring in simple graphs and many results in list coloring can be generalized in those of DP-coloring. In this work, we introduce a relaxed DP-coloring which is a generalization if a relaxed list coloring. We also shows that every planar graph $G$ without $4$-cycles or $6$-cycles is DP-$(k,d)^*$-colorable. It follows immediately that $G$ is $(k,d)^*$-choosable.
A graph $G$ is $(d_1,d_2,d_3)$-colorable if the vertex set $V(G)$ can be partitioned into three subsets $V_1,V_2$ and $V_3$ such that for $i\in\{1,2,3\}$, the induced graph $G[V_i]$ has maximum vertex-degree at most $d_i$. So, $(0,0,0)$-colorability is exactly 3-colorability. The well-known Steinberg's conjecture states that every planar graph without cycles of length 4 or 5 is 3-colorable. As this conjecture being disproved by Cohen-Addad etc. in 2017, a similar question, whether every planar graph without cycles of length 4 or $i$ is 3-colorable for a given $i\in \{6,\ldots,9\}$, is gaining …
Aksenov proved that in a planar graph G with at most one triangle, every precoloring of a 4-cycle can be extended to a 3-coloring of G. We give an exact characterization of planar graphs with two triangles in that some precoloring of a 4-cycle does not extend. We apply this characterization to solve the precoloring extension problem from two 4-cycles in a triangle-free planar graph in the case that the precolored 4-cycles are separated by many disjoint 4-cycles. The latter result is used in followup papers to give detailed information about the structure of 4-critical triangle-free graphs embe…
DP-coloring of a simple graph is a generalization of list coloring, and also a generalization of signed coloring of signed graphs. It is known that for each $k \in \{3, 4, 5, 6\}$, every planar graph without $C_k$ is 4-choosable. Furthermore, Jin, Kang, and Steffen \cite{JKS} showed that for each $k \in \{3, 4, 5, 6\}$, every signed planar graph without $C_k$ is signed 4-choosable. In this paper, we show that for each $k \in \{3, 4, 5, 6\}$, every planar graph without $C_k$ is 4-DP-colorable, which is an extension of the above results.
We show, without using the Four Color Theorem, that for each planar triangulation, the number of its proper vertex colorings by 4 colors is a determinant and thus can be calculated in a polynomial time. In particular, we can efficiently decide if the number is non-zero.
We derive sharp upper and lower bounds on the number of intersection points and closed regions that can occur in sets of line segments with certain structure, in terms of the number of segments. We consider sets of segments whose underlying planar graphs are Halin graphs, cactus graphs, maximal planar graphs, and triangle-free planar graphs, as well as randomly produced segment sets. We also apply these results to a variant of the Erdős-Faber-Lovász (EFL) Conjecture stating that the intersection points of $m$ segments can be colored with $m$ colors so that no segment contains points with the s…
In this paper, an algorithm for determining 3-colorability, i.e. the decision problem (YES/NO), in planar graphs is presented. The algorithm, although not exact (it could produce false positives) has two very important features: (i) it has polynomial complexity and (ii) for every "NO" answer, a "short" proof is generated, which is of much interest since 3-colorability is a NP-complete problem and thus its complementary problem is in Co-NP. Hence the algorithm is exact when it determines that a given planar graph is not 3-colorable since this is verifiable via an automatic generation of short f…
It is shown that for any fixed $c \geq 3$ and $r$, the maximum possible chromatic number of a graph on $n$ vertices in which every subgraph of radius at most $r$ is $c$ colorable is $\tildeΘ\left(n ^ {\frac{1}{r+1}} \right)$ (that is, $n^\frac{1}{r+1}$ up to a factor poly-logarithmic in $n$). The proof is based on a careful analysis of the local and global colorability of random graphs and implies, in particular, that a random $n$-vertex graph with the right edge probability has typically a chromatic number as above and yet most balls of radius $r$ in it are $2$-degenerate.
L.M.Vitaver [1962] and G.I.Minty [1962] suggested criteria for vertex colorability of a graph in at most a given number of colors; these criteria are stated in terms of the orientation of the edges. One additional criterion of this kind is given here.
Let $G$ be an $n$-vertex graph with the maximum degree $Δ$ and the minimum degree $δ$. We give algorithms with complexity $O(1.3158^{n-0.7~Δ(G)})$ and $O(1.32^{n-0.73~Δ(G)})$ that determines if $G$ is 3-colorable, when $δ(G)\geq 8$ and $δ(G)\geq 7$, respectively.
The series-parallel (SP) graphs are those containing no topological $K_{_4}$ and are considered trivial. We relax the prohibition distinguishing the SP graphs by forbidding only embeddings of $K_{_4}$ whose edges with both ends 3-valent (skeleton hereafter) induce a graph isomorphic to certain prescribed subgraphs of $K_{_4}$. In particular, we describe the structure of the graphs containing no embedding of $K_{_4}$ whose skeleton is isomorphic to $P_{_3}$ or $P_{_4}$. Such "almost series-parallel graphs" (ASP) still admit a concise description. Amongst other things, their description reveals …
Stan Wagon asked the following in 2000. Is every zonohedron face 3-colorable when viewed as a planar map? An equivalent question, under a different guise, is the following: is the arrangement graph of great circles on the sphere always vertex 3-colorable? (The arrangement graph has a vertex for each intersection point, and an edge for each arc directly connecting two intersection points.) Assume that no three circles meet at a point, so that this arrangement graph is 4-regular. In this note we have shown that all arrangement graphs defined as above are 3-colorable.
Let $G$ be an $n$-vertex graph and let $L:V(G)\rightarrow P(\{1,2,3\})$ be a list assignment over the vertices of $G$, where each vertex with list of size 3 and of degree at most 5 has at least three neighbors with lists of size 2. We can determine $L$-choosability of $G$ in $O(1.3196^{n_3+.5n_2})$ time, where $n_i$ is the number of vertices in $G$ with list of size $i$ for $i\in \{2,3\}$. As a corollary, we conclude that the 3-colorability of any graph $G$ with minimum degree at least 6 can be determined in $O(1.3196^{n-.5Δ(G)})$ time.
A graph G is list (b:a)-colorable if for every assignment of lists of size b to vertices of G, there exists a choice of an a-element subset of the list at each vertex such that the subsets chosen at adjacent vertices are disjoint. We prove that for every positive integer a, the family of minimal obstructions of girth at least five to list (3a:a)-colorability is strongly hyperbolic, in the sense of the hyperbolicity theory developed by Postle and Thomas. This has a number of consequences, e.g., that if a graph of girth at least five and Euler genus g is not list (3a:a)-colorable, then G contain…
We give an elementary proof for the non-$3$-colorability of $4$-chromatic Schrijver graphs thus providing such a proof also for $4$-chromatic Kneser graphs. To this end we use a complete description of the structure of $4$-chromatic Schrijver graphs that was already given by Braun and even earlier in an unpublished manuscript by Li. We also address connections to surface quadrangulations. In particular, we show that a spanning subgraph of $4$-chromatic Schrijver graphs quadrangulates the Klein bottle, while another spanning subgraph quadrangulates the projective plane. The latter is a special …
We give a linear-time algorithm to decide 3-colorability (and find a 3-coloring, if it exists) of quadrangulations of a fixed surface. The algorithm also allows to prescribe the coloring for a bounded number of vertices.
In a seminal paper, De Loera et. al introduce the algorithm NulLA (Nullstellensatz Linear Algebra) and use it to measure the difficulty of determining if a graph is not 3-colorable. The crux of this relies on a correspondence between 3-colorings of a graph and solutions to a certain system of polynomial equations over a field $\mathbb{k}$. In this article, we give a new direct combinatorial characterization of graphs that can be determined to be non-3-colorable in the first iteration of this algorithm when $\mathbb{k}=GF(2)$. This greatly simplifies the work of De Loera et. al, as we express t…
We confirm the equitable $Δ$-coloring conjecture for interval graphs and establish the monotonicity of equitable colorability for them. We further obtain results on equitable colorability about square (or Cartesian) and cross (or direct) products of graphs.
A word $\bar{w} = \bar{u}\bar{u}$ is a $long$ $square$ if $\bar{u}$ is of length at least 3; a word $\bar{w}$ is $long$-$square$-$free$ if $\bar{w}$ contains no sub-word that is a long square. We can use words to generate graph colorings; a graph coloring is called $long$-$repetition$-$free$ if the word formed by the coloring of each path in the graph is long-square-free. Our results show that every rooted tree of radius less than or equal to seven is long-repetition-free two-colorable. We also prove there exists a class of trees which are not long-repetition-free two-colorable.
For a graph G, let h(G) denote the largest k such that G has k pairwise disjoint pairwise adjacent connected nonempty subgraphs, and let s(G) denote the largest k such that G has k pairwise disjoint pairwise adjacent connected subgraphs of size 1 or 2. Hadwiger's conjecture states that h(G) is at least c(G), where c(G) is the chromatic number of G. Seymour conjectured that s(G) is at least |V(G)|/2 for all graphs without antitriangles, i. e. three pairwise nonadjacent vertices. Here we concentrate on graphs G with exactly one c(G)-coloring. We prove generalizations of (i) if c(G) is at most 6 …
We give an exact characterization of 3-colorability of triangle-free graphs drawn in the torus, in the form of 186 "templates" (graphs with certain faces filled by arbitrary quadrangulations) such that a graph from this class is not 3-colorable if and only if it contains a subgraph matching one of the templates. As a consequence, we show every triangle-free graph drawn in the torus with edge-width at least six is 3-colorable, a key property used in an efficient 3-colorability algorithm for triangle-free toroidal graphs.
The incidence coloring conjecture, proposed by Brualdi and Massey in 1993, states that the incidence coloring number of every graph is at most ${\it Δ}+2$, where ${\it Δ}$ is the maximum degree of a graph. The conjecture was shown to be false in general by Guiduli in 1997, following the work of Algor and Alon. However, in 2005 Maydanskiy proved that the conjecture holds for any graph with ${\it Δ}\leq 3$. It is easily deduced that the incidence coloring number of a semi-cubic graph is 4 or 5. In this paper, we show that it is already NP-complete to determine if a semi-cubic graph is 4-incidenc…
We characterize $k$--colorability of a simplicial graph via the intrinsic algebraic structure of the associated right-angled Artin group. As a consequence, we show that a certain problem about the existence of homomorphisms from right-angled Artin groups to products of free groups is NP--complete.
We consider the following two-player game: Maxi and Mini start with the empty graph on $n$ vertices and take turns, always adding one additional edge to the graph such that the chromatic number is at most $k$, where $k \in \mathbb{N}$ is a given parameter. The game is over when the graph is saturated and no further edge can be inserted. Maxi wants to maximize the length of the game while Mini wants to minimize it. The score $s(n,χ_{>k})$ denotes the number of edges in the final graph, given that both players followed an optimal strategy. This colorability game belongs to the family of \emph{sa…
A graph is k-choosable if it can be colored whenever every vertex has a list of at least k available colors. We prove that if cycles of length at most four in a planar graph G are pairwise far apart, then G is 3-choosable. This is analogous to the problem of Havel regarding 3-colorability of planar graphs with triangles far apart.
The study of graph vertex colorability from an algebraic perspective has introduced novel techniques and algorithms into the field. For instance, it is known that $k$-colorability of a graph $G$ is equivalent to the condition $1 \in I_{G,k}$ for a certain ideal $I_{G,k} \subseteq \k[x_1, ..., x_n]$. In this paper, we extend this result by proving a general decomposition theorem for $I_{G,k}$. This theorem allows us to give an algebraic characterization of uniquely $k$-colorable graphs. Our results also give algorithms for testing unique colorability. As an application, we verify a counterexamp…
An injective coloring of a graph $G$ is an assignment of colors to the vertices of $G$ so that any two vertices with a common neighbor have distinct colors. A graph $G$ is injectively $k$-choosable if for any list assignment $L$, where $|L(v)| \geq k$ for all $v \in V(G)$, $G$ has an injective $L$-coloring. Injective colorings have applications in the theory of error-correcting codes and are closely related to other notions of colorability. In this paper, we show that subcubic planar graphs with girth at least 6 are injectively 5-choosable. This strengthens a result of Lužar, Škrekovski, and T…
In this paper we have given a unified graph coloring algorithm for planar graphs. The problems that have been considered in this context respectively, are vertex, edge, total and entire colorings of the planar graphs. The main tool in the coloring algorithm is the use of spiral chain which has been used in the non-computer proof of the four color theorem in 2004. A more precies explanation of the proof of the four color theorem by spiral chain coloring is also given in this paper. Then we continue to spiral-chain coloring solutions by giving the proof of other famous conjectures of Vizing's to…
If each edge (u,v) of a graph G=(V,E) is decorated with a permutation pi_{u,v} of k objects, we say that it has a permuted k-coloring if there is a coloring sigma from V to {1,...,k} such that sigma(v) is different from pi_{u,v}(sigma(u)) for all (u,v) in E. Based on arguments from statistical physics, we conjecture that the threshold d_k for permuted k-colorability in random graphs G(n,m=dn/2), where the permutations on the edges are uniformly random, is equal to the threshold for standard graph k-colorability. The additional symmetry provided by random permutations makes it easier to prove b…
We study the computational efficiency of approaches, based on Hilbert's Nullstellensatz, which use systems of linear equations for detecting non-colorability of graphs having large girth and chromatic number. We show that for every non-$k$-colorable graph with $n$ vertices and girth $g>4k$, the algorithm is required to solve systems of size at least $n^{Ω(g)}$ in order to detect its non-$k$-colorability.
The Grötzsch Theorem states that every triangle-free planar graph admits a proper $3$-coloring. Among many of its generalizations, the one of Grünbaum and Aksenov, giving $3$-colorability of planar graphs with at most three triangles, is perhaps the most known. A lot of attention was also given to extending $3$-colorings of subgraphs to the whole graph. In this paper, we consider $3$-colorings of planar graphs with at most one triangle. Particularly, we show that precoloring of any two non-adjacent vertices and precoloring of a face of length at most $4$ can be extended to a $3$-coloring of th…
We study the problem of efficiently refuting the k-colorability of a graph, or equivalently certifying a lower bound on its chromatic number. We give formal evidence of average-case computational hardness for this problem in sparse random regular graphs, showing optimality of a simple spectral certificate. This evidence takes the form of a computationally-quiet planting: we construct a distribution of d-regular graphs that has significantly smaller chromatic number than a typical regular graph drawn uniformly at random, while providing evidence that these two distributions are indistinguishabl…
Let $P_t$ and $C_\ell$ denote a path on $t$ vertices and a cycle on $\ell$ vertices, respectively. In this paper we study the $k$-coloring problem for $(P_t,C_\ell)$-free graphs. Maffray and Morel, and Bruce, Hoang and Sawada, have proved that 3-colorability of $P_5$-free graphs has a finite forbidden induced subgraphs characterization, while Hoang, Moore, Recoskie, Sawada, and Vatshelle have shown that $k$-colorability of $P_5$-free graphs for $k \geq 4$ does not. These authors have also shown, aided by a computer search, that 4-colorability of $(P_5,C_5)$-free graphs does have a finite forbi…
A total $k$-coloring of a graph is an assignment of $k$ colors to its vertices and edges such that no two adjacent or incident elements receive the same color. The Total Coloring Conjecture (TCC) states that every simple graph $G$ has a total ($Δ(G)+2$)-coloring, where $Δ(G)$ is the maximum degree of $G$. This conjecture has been confirmed for planar graphs with maximum degree at least 7 or at most 5, i.e., the only open case of TCC is that of maximum degree 6. It is known that every planar graph $G$ of $Δ(G) \geq 9$ or $Δ(G) \in \{7, 8\}$ with some restrictions has a total $(Δ(G) + 1)$-colori…
In spite of the extensive studies of the 3-coloring problem with respect to several basic parameters, the complexity status of the 3-coloring problem on graphs with small diameter, i.e. with diameter 2 or 3, has been a longstanding and challenging open question. For graphs with diameter 2 we provide the first subexponential algorithm with complexity $2^{O(\sqrt{n\log n})}$, which is asymptotically the same as the currently best known time complexity for the graph isomorphism (GI) problem. Moreover, we prove that the graph isomorphism problem on 3-colorable graphs with diameter 2 is GI-complete…
Despite the existence of a proof of the 4-color theorem, it would seem that there is still more to learn about why any planar graph is 4-colorable. To that end, we take another look at the Birkhoff diamond and discover something new and intriguing: after an extensive search for (rare) Kempe-locked triangulations, we find a Birkhoff diamond subgraph in each one. We offer a heuristic argument as to why that result is not only reasonable but also to be expected and posit that the presence of a Birkhoff diamond is necessary to Kempe-locking. If that conjecture is true, it means that the Birkhoff d…
We give a linear-time algorithm to decide 3-colorability of a triangle-free graph embedded in a fixed surface, and a quadratic-time algorithm to output a 3-coloring in the affirmative case. The algorithms also allow to prescribe the coloring of a bounded number of vertices.
A graph is $H$-free if it has no induced subgraph isomorphic to $H$. We characterize all graphs $H$ for which there are only finitely many minimal non-three-colorable $H$-free graphs. Such a characterization was previously known only in the case when $H$ is connected. This solves a problem posed by Golovach et al. As a second result, we characterize all graphs $H$ for which there are only finitely many $H$-free minimal obstructions for list 3-colorability.
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…
Weak degeneracy is a variation of degeneracy which shares many nice properties of degeneracy. In particular, if a graph $G$ is weakly $d$-degenerate, then for any $(d + 1)$-list assignment $L$ of $G$, one can construct an $L$-coloring of $G$ by a modified greedy coloring algorithm. It is known that planar graphs of girth 5 are 3-choosable and locally planar graphs are 5-choosable. This paper strengthens these results and proves that planar graphs of girth 5 are weakly 2-degenerate and locally planar graphs are weakly 4-degenerate.
For a positive integer $k$, the $k$-recolouring graph of a graph $G$ has as vertex set all proper $k$-colourings of $G$ with two $k$-colourings being adjacent if they differ by the colour of exactly one vertex. A result of Dyer et al. regarding graphs of bounded degeneracy implies that the $7$-recolouring graphs of planar graphs, the $5$-recolouring graphs of triangle-free planar graphs and the $4$-recolouring graphs planar graphs of girth at least six are connected. On the other hand, there are planar graphs whose $6$-recolouring graph is disconnected, triangle-free planar graphs whose $4$-re…
