arXiv subjects

Reza Naserasr

Publications and source records attributed to Reza Naserasr.

At least 19 recordsLinked to original sources

arXiv2026

Winding number and circular coloring

In 1996, Youngs proved a surprising theorem that quadrangulations of the projective plane could never have chromatic number exactly 3. This sparked a lot of interest, and the result has been further developed in many directions over the past decades. For example, the result is strengthened by considering the circular chromatic number, which is a real-valued lower bound on the chromatic number. The circular chromatic number of a quadrangulation cannot be in the interval (2,4). This parameter allows a generalization to larger even faces, for which a similar gap exists. In this work, we place these results into a framework based on the notion of winding number using extensions of colorings to continuous mappings. This yields unified and simplified proofs of gaps in the circular chromatic number for graphs with a distinguished set of directed even cycles. This generalizes the setting of graphs embedded on surfaces where every face is even. We further establish an analogous gap phenomenon when all faces are of a given odd length, previously known only in the case of triangulations. For example, we conclude that if G is a graph embedded on the projective plane such that all faces are 5-cycles, then either its circular chromatic number is 5/2 or at least 3, the former being the case only if G is Eulerian and every noncontractible facial walk is of odd length...

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arXiv2025

Bounding signed bipartite partial t-trees and application to edge-coloring

Given a signed bipartite graph $(B, \pi)$ of negative girth $2k$, we present a necessary and sufficient condition for it to have the following property: each signed bipartite graph $(G, \sigma)$ whose negative girth is at least $2k$ and whose underlying graph has treewidth at most $t$ admits a homomorphism to $(B, \pi)$. Applying the result on the signed projective cube $SPC(2k-1)$, we conclude that every signed bipartite graph of negative girth at least $2k$ whose underlying graph is a partial 3-tree admits a homomorphism to $SPC(2k-1)$. For planar partial 3-trees, applying duality we conclude that if $G$ is a planar $2k$-regular multigraph whose dual has treewidth at most 3 and such that every edge-cut $(X, V\backslash X)$, where $|X|$ is odd, has size at least $2k$, then $G$ is $2k$-edge-colorable. This supports a conjecture of Seymour which, in full generality, largely extends Tait's reformulation of the four-color theorem, claiming that the fractional edge-chromatic number of a planar multigraph determines its edge-chromatic number. Finally, noting the contrast between fractional isomorphism and quantum isomorphism, where the former admits a polynomial time algorithm while the latter is proved to be undecidable, and observing the similarities of these notions to the subject of our study, we ask if there is an algorithm to decide if an input signed graph $\widehat{B}$ has the following property: if a signed planar graph $\widehat{G}$ does not map to $\widehat{B}$, it would be because a cycle in $\widehat{G}$ does not map to $\widehat{B}$. In other words, minimal planar graphs that do not map to $\widehat{B}$ are signed cycles.

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arXiv2025

Extension of the Gy\'arf\'as-Sumner conjecture to signed graphs

The balanced chromatic number of a signed graph G is the minimum number of balanced sets that cover all vertices of G. Studying structural conditions which imply bounds on the balanced chromatic number of signed graphs is among the most fundamental problems in graph theory. In this work, we initiate the study of coloring hereditary classes of signed graphs. More precisely, we say that a set F = {F_1, F_2, ..., F_l} is a GS (for Gy\'arf\'as-Sumner) set if there exists a constant c such that signed graphs with no induced subgraph switching equivalent to a member of F admit a balanced c-coloring. The focus of this work is to study GS sets of order 2. We show that if F is a GS set of order 2, then F_1 is either (K_3, -) or (K_4, -), and F_2 is a linear forest. In the case of F_1 = (K_3, -), we show that any choice of a linear forest for F_2 works. In the case of F_1 = (K_4, -), we show that if each connected component of F_2 is a path of length at most 4, then {F_1, F_2} is a GS set.

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arXiv2025

Brooks' theorem for signed graphs with $Δ=3$

Circular $r$-coloring of a signed graph $(G,σ)$ is a mapping of its vertices to a circle of circumference $r$ such that: I. each pair of vertices with a negative connection is at distance at least $1$, and II. for each pair with a positive connection, the distance of one from the antipodal of the other is at least $1$. A signed graph $(G,σ)$ admits a circular $r$-coloring for some values of $r$ if and only if it has no negative loop. The smallest value of such $r$ is the circular chromatic number, denoted $χ_{c}(G,σ)$. The circular chromatic number is a refinement of the balanced chromatic number, which is mostly studied under the equivalent term $0$-free coloring in the literature. Extending Brooks' theorem, Má\v cajová, Raspaud, and Škoviera showed that if $Δ(G)$ is an even number, $G$ is connected, and $(G,σ)$ is not (switching) isomorphic to $(K_{Δ+1},-)$ or $C_{-\ell}$ (when $Δ(G)=2$), then $χ_c(G,σ)\leq Δ(G)$ and that the upper bound is tight. For the odd values of $Δ(G)$, assuming a connected signed graph $(G,σ)$ is not isomorphic to $(K_{Δ+1},-)$, determining the best upper bound for $χ_c(G, σ)$ proves to be more of a challenge. In this work, addressing the first step of this question, we show that if $(G, σ)$ is a signed graph of maximum degree 3 with no component isomorphic to $(K_4, -)$, then $χ_{c}(G, σ)\leq \frac{10}{3}$. The upper bound is tight even among signed cubic graphs of girth 5. In particular, there is a signature on the Petersen graph for which the upper of $\frac{10}{3}$ is achieved.

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arXiv2025

Colouring signed analogues of Kneser, Schrijver, and Borsuk graphs

The Kneser signed graph $\KS(n,k)$, $k\leq n$, is the graph whose vertices are signed $k$-subsets of $[n]$ (i.e. $k$-subsets $S$ of $\{ \pm 1, \pm 2, \ldots, \pm n\}$ such that $S\cap (-S)=\emptyset$). Two vertices $A$ and $B$ are adjacent with a positive edge if $A\cap (-B)=\emptyset$ and with a negative edge if $A\cap B=\emptyset$. We prove that the balanced chromatic number of $\KS(n,k)$ is $n-k+1$. We then introduce the signed analogue of Schrijver graphs and show that they form vertex-critical subgraphs of $\KS(n,k)$ with respect to balanced colouring. Further connection to topological methods, in particular, connection to Borsuk signed graphs is also considered.

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arXiv2025

Fractional balanced chromatic number and arboricity of planar (signed) graphs

A fractional coloring of a signed graph $(G, σ)$ is an assignment of nonnegative weights to the balanced sets (sets which do not induce a negative cycle) such that each vertex has an accumulated weight of at least 1. The minimum total wight among all such colorings is defined to be the fractional balanced chromatic number, denoted by $χ-{fb}(G, σ)$. This value is clearly upper bounded by the fractional arboricity of $G$, denoted $a_f (G)$, where weights are assigned to sets inducing no cycle rather than sets inducing no negative cycle. In this work we present an example of a planar signed simple graph of fractional balanced chromatic number larger than 2, thus in particular refuting a conjecture of Bonamy, Kardos, Kelly, and Postle suggesting that the fractional arboricity of planar graphs is bounded above by 2. By iterating the construction, we show that the supremum of the fractional balanced chromatic number of planar signed simple graphs is at least as $83/41 = 2 + 1/41$. With similar operations, we built a sequence of planar graphs whose limit of fractional arboricity is $a_f (G) = 2 + 2/25$.

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arXiv2025

Odd Hadwiger's conjecture for the complements of Kneser graphs

A generalization of the four-color theorem, Hadwiger's conjecture is considered as one of the most important and challenging problems in graph theory, and odd Hadwiger's conjecture is a strengthening of Hadwiger's conjecture by way of signed graphs. In this paper, we prove that odd Hadwiger's conjecture is true for the complements $\overline{K}(n,k)$ of the Kneser graphs $K(n,k)$, where $n\geq 2k \ge 4$. This improves a result of G. Xu and S. Zhou (2017) which states that Hadwiger's conjecture is true for this family of graphs. Moreover, we prove that $\overline{K}(n,k)$ contains a 1-shallow complete minor of a special type with order no less than the chromatic number $χ(\overline{K}(n,k))$, and in the case when $7 \le 2k+1 \le n \le 3k-1$ the gap between the odd Hadwiger number and chromatic number of $\overline{K}(n,k)$ is $Ω(1.5^{k})$.

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arXiv2025

On core of categorical product of (di)graphs

The core of a graph is the smallest graph (in terms of number of vertices) to which it is homomorphically equivalent. The question of the possible order of the core of the tensor product (also known as categorical, Heidetnemi or direct product) of two graphs captures some well known problems. For instance, the recent counterexample to the Hedetniemi conjecture for 5-chromatic graphs is equivalent to saying that there are cores of order at least 5 whose product has a core of order 4. In this work, motivated by a question from Leonid Libkin in the area of graph databases, we first present methods of building cores whose categorical product is also a core. Extending on this we present sufficient conditions for a set of cores to have a product which is also a core. Presenting an example of such a family of digraphs, we construct a family of $\binom{2n}{n}$ digraphs, where the number of vertices of each is between $n^2+5n+2$ and $3n^2+3n+2$ and the product is a core. We then present a method of transforming the example into a family of graphs.

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arXiv2025

Fractional balanced chromatic number of signed subcubic graphs

A signed graph is a pair $(G,σ)$, where $G$ is a graph and $σ: E(G)\rightarrow \{-, +\}$, called signature, is an assignment of signs to the edges. Given a signed graph $(G,σ)$ with no negative loops, a balanced $(p,q)$-coloring of $(G,σ)$ is an assignment $f$ of $q$ colors to each vertex from a pool of $p$ colors such that each color class induces a balanced subgraph, i.e., no negative cycles. Let $(K_4,-)$ be the signed graph on $K_4$ with all edges being negative. In this work, we show that every signed (simple) subcubic graph admits a balanced $(5,3)$-coloring except for $(K_4,-)$ and signed graphs switching equivalent to it. For this particular signed graph the best balanced colorings are $(2p,p)$-colorings.

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arXiv2024

Colouring negative exact-distance graphs of signed graphs

The $k$-th exact-distance graph, of a graph $G$ has $V(G)$ as its vertex set, and $xy$ as an edge if and only if the distance between $x$ and $y$ is (exactly) $k$ in $G$. We consider two possible extensions of this notion for signed graphs. Finding the chromatic number of a negative exact-distance square of a signed graph is a weakening of the problem of finding the smallest target graph to which the signed graph has a sign-preserving homomorphism. We study the chromatic number of negative exact-distance graphs of signed graphs that are planar, and also the relation of these chromatic numbers with the generalised colouring numbers of the underlying graphs. Our results are related to a theorem of Alon and Marshall about homomorphisms of signed graphs.

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arXiv2024

Signed projective cubes, a homomorphism point of view

The (signed) projective cubes, as a special class of graphs closely related to the hypercubes, are on the crossroad of geometry, algebra, discrete mathematics and linear algebra. Defined as Cayley graphs on binary groups, they represent basic linear dependencies. Capturing the four-color theorem as a homomorphism target they show how mapping of discrete objects, namely graphs, may relate to special mappings of plane to projective spaces of higher dimensions. In this work, viewed as a signed graph, first we present a number of equivalent definitions each of which leads to a different development. In particular, the new notion of common product of signed graphs is introduced which captures both Cartesian and tensor products of graphs. We then have a look at some of their homomorphism properties. We first introduce an inverse technique for the basic no-homomorphism lemma, using which we show that every signed projective cube is of circular chromatic number 4. Then observing that the 4-color theorem is about mapping planar graphs into signed projective cube of dimension 2, we study some conjectures in extension of 4CT. Toward a better understanding of these conjectures we present the notion of extended double cover as a key operation in formulating the conjectures. With a deeper look into connection between some of these graphs and algebraic geometry, we discover that projective cube of dimension 4, widely known as the Clebsh graph, but also known as Greenwood-Gleason graph, is the intersection graph of the 16 straight lines of an algebraic surface known as Segre surface, which is a Del Pezzo surface of degree 4. We note that an algebraic surface known as the Clebsch surface is one of the most symmetric presentations of a cubic surface. Recall that each smooth cubic surface contains 27 lines. Hence, from hereafter, we believe, a proper name for this graph should be Segre graph.

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arXiv2024

Winding number and circular 4-coloring of signed graphs

Concerning the recent notion of circular chromatic number of signed graphs, for each given integer $k$ we introduce two signed bipartite graphs, each on $2k^2-k+1$ vertices, having shortest negative cycle of length $2k$, and the circular chromatic number 4. Each of the construction can be viewed as a bipartite analogue of the generalized Mycielski graphs on odd cycles, $M_{\ell}(C_{2k+1})$. In the course of proving our result, we also obtain a simple proof of the fact that $M_{\ell}(C_{2k+1})$ and some similar quadrangulations of the projective plane have circular chromatic number 4. These proofs have the advantage that they illuminate, in an elementary manner, the strong relation between algebraic topology and graph coloring problems.

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arXiv2023

Balanced-chromatic number and Hadwiger-like conjectures

Motivated by different characterizations of planar graphs and the 4-Color Theorem, several structural results concerning graphs of high chromatic number have been obtained. Toward strengthening some of these results, we consider the \emph{balanced chromatic number}, $\chi_b(\hat{G})$, of a signed graph $\hat{G}$. This is the minimum number of parts into which the vertices of a signed graph can be partitioned so that none of the parts induces a negative cycle. This extends the notion of the chromatic number of a graph since $\chi(G)=\chi_b(\tilde{G})$, where $\tilde{G}$ denotes the signed graph obtained from~$G$ by replacing each edge with a pair of (parallel) positive and negative edges. We introduce a signed version of Hadwiger's conjecture as follows. Conjecture: If a signed graph $\hat{G}$ has no negative loop and no $\tilde{K_t}$-minor, then its balanced chromatic number is at most $t-1$. We prove that this conjecture is, in fact, equivalent to Hadwiger's conjecture and show its relation to the Odd Hadwiger Conjecture. Motivated by these results, we also consider the relation between subdivisions and balanced chromatic number. We prove that if $(G, \sigma)$ has no negative loop and no $\tilde{K_t}$-subdivision, then it admits a balanced $\frac{79}{2}t^2$-coloring. This qualitatively generalizes a result of Kawarabayashi (2013) on totally odd subdivisions.

math.CO↗

arXiv2023

Characterizing and recognizing exact-distance squares of graphs

For a graph $G=(V,E)$, its exact-distance square, $G^{[\sharp 2]}$, is the graph with vertex set $V$ and with an edge between vertices $x$ and $y$ if and only if $x$ and $y$ have distance (exactly) $2$ in $G$. The graph $G$ is an exact-distance square root of $G^{[\sharp 2]}$. We give a characterization of graphs having an exact-distance square root, our characterization easily leading to a polynomial-time recognition algorithm. We show that it is NP-complete to recognize graphs with a bipartite exact-distance square root. These two results strongly contrast known results on (usual) graph squares. We then characterize graphs having a tree as an exact-distance square root, and from this obtain a polynomial-time recognition algorithm for these graphs. Finally, we show that, unlike for usual square roots, a graph might have (arbitrarily many) non-isomorphic exact-distance square roots which are trees.

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arXiv2023

Critically 3-frustrated signed graphs

Extending the notion of maxcut, the study of the frustration index of signed graphs is one of the basic questions in the theory of signed graphs. Recently two of the authors initiated the study of critically frustrated signed graphs. That is a signed graph whose frustration index decreases with the removal of any edge. The main focus of this study is on critical signed graphs which are not edge-disjoint unions of critically frustrated signed graphs (namely non-decomposable signed graphs) and which are not built from other critically frustrated signed graphs by subdivision. We conjecture that for any given $k$ there are only finitely many critically $k$-frustrated signed graphs of this kind. Providing support for this conjecture we show that there are only two of such critically $3$-frustrated signed graphs where there is no pair of edge-disjoint negative cycles. Similarly, we show that there are exactly ten critically $3$-frustrated signed planar graphs that are neither decomposable nor subdivisions of other critically frustrated signed graphs. We present a method for building non-decomposable critically frustrated signed graphs based on two given such signed graphs. We also show that the condition of being non-decomposable is necessary for our conjecture.

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arXiv2022

Circular flows in mono-directed signed graphs

In this paper the concept of circular $r$-flows in a mono-directed signed graph $(G, σ)$ is introduced. That is a pair $(D, f)$, where $D$ is an orientation on $G$ and $f: E(G)\to (-r,r)$ satisfies that $|f(e)|\in [1, r-1]$ for each positive edge $e$ and $|f(e)|\in [0, \frac{r}{2}-1]\cup [\frac{r}{2}+1, r)$ for each negative edge $e$, and the total in-flow equals the total out-flow at each vertex. The circular flow index of a signed graph $(G, σ)$ with no positive bridge, denoted $Φ_c(G,σ)$, is the minimum $r$ such that $(G, σ)$ admits a circular $r$-flow. This is the dual notion of circular colorings and circular chromatic numbers of signed graphs recently introduced in [Circular chromatic number of signed graphs. R. Naserasr, Z. Wang, and X. Zhu. Electronic Journal of Combinatorics, 28(2)(2021), \#P2.44], and is distinct from the concept of circular flows in bi-directed graphs associated to signed graphs studied in the literature. We give several equivalent definitions, study basic properties of circular flows in mono-directed signed graphs, explore relations with flows in graphs, and focus on upper bounds on $Φ_c(G,σ)$ in terms of the edge-connectivity of $G$. Meanwhile, we note that for the particular values of $r_{_k}=\frac{2k}{k-1}$, and when restricted to two natural subclasses of signed graphs, the existence of a circular $r_{_k} $-flow is strongly connected with the existence of a modulo $k$-orientation, and in case of planar graphs, based on duality, with the homomorphisms to $C_{-k}$.

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arXiv2022

A New Graph Parameter To Measure Linearity

Consider a sequence of LexBFS vertex orderings σ1, σ2, . . . where each ordering σi is used to break ties for σi+1. Since the total number of vertex orderings of a finite graph is finite, this sequence must end in a cycle of vertex orderings. The possible length of this cycle is the main subject of this work. Intuitively, we prove for graphs with a known notion of linearity (e.g., interval graphs with their interval representation on the real line), this cycle cannot be too big, no matter which vertex ordering we start with. More precisely, it was conjectured in [9] that for cocomparability graphs, the size of this cycle is always 2, independent of the starting order. Furthermore [27] asked whether for arbitrary graphs, the size of such a cycle is always bounded by the asteroidal number of the graph. In this work, while we answer this latter question negatively, we provide support for the conjecture on cocomparability graphs by proving it for the subclass of domino-free cocomparability graphs. This subclass contains cographs, proper interval, interval, and cobipartite graphs. We also provide simpler independent proofs for each of these cases which lead to stronger results on this subclasses.

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arXiv2022

Smallest $C_{2l+1}$-critical graphs of odd-girth $2k+1$

Given a graph $H$, a graph $G$ is called $H$-critical if $G$ does not admit a homomorphism to $H$, but any proper subgraph of $G$ does. Observe that $K_{k-1}$-critical graphs are the standard $k$-(colour)-critical graphs. We consider questions of extremal nature previously studied for $k$-critical graphs and generalize them to $H$-critical graphs. After complete graphs, the next natural case to consider for $H$ is that of the odd-cycles. Thus, given integers $\ell$ and $k$, $\ell\geq k$, we ask: what is the smallest order of a $C_{2\ell +1}$-critical graph of odd-girth at least $2k+1$? Denoting this value by $η(k,C_{2\ell+1})$, we show that $η(k,C_{2\ell+1})=4k$ for $1\leq\ell\leq k\leq\frac{3\ell+i-3}{2}$ ($2k=i\bmod 3$) and that $η(3,C_5)=15$. The latter means that a smallest graph of odd-girth~$7$ not admitting a homomorphism to the $5$-cycle is of order~$15$. Computational work shows that there are exactly eleven such graphs on $15$~vertices of which only two are $C_5$-critical.

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