arXiv subjects

Bojan Mohar

Publications and source records attributed to Bojan Mohar.

At least 37 records · Page 2Linked to original sources

arXiv2025

Hermitian adjacency matrices with at most three distinct eigenvalues

We study oriented graphs whose Hermitian adjacency matrices of the second kind have few eigenvalues. We give a complete characterization of the oriented graphs with two distinct eigenvalues, showing that there are only four such graphs. We extend this result to mixed graphs. We show that there are infinitely many regular tournaments with three distinct eigenvalues. We extend our main results to Hermitian adjacency matrices defined over other roots of unity.

math.CO↗

arXiv2025

Sharing tea on a graph

Motivated by the analysis of consensus formation in the Deffuant model for social interaction, we consider the following procedure on a graph $G$. Initially, there is one unit of tea at a fixed vertex $r \in V(G)$, and all other vertices have no tea. At any time in the procedure, we can choose a connected subset of vertices $T$ and equalize the amount of tea among vertices in $T$. We prove that if $x \in V(G)$ is at distance $d$ from $r$, then $x$ will have at most $\frac{1}{d+1}$ units of tea during any step of the procedure. This bound is best possible and answers a question of Gantert. We also consider arbitrary initial weight distributions. For every finite graph $G$ and $w \in \mathbb{R}_{\geq 0}^{V(G)}$, we prove that the set of weight distributions reachable from $w$ is a compact subset of $\mathbb{R}_{\geq 0}^{V(G)}$.

math.CO↗

arXiv2025

Vertex Partitioning and $p$-Energy of Graphs

For a Hermitian matrix $A$ of order $n$ with eigenvalues $λ_1(A)\ge \cdots\ge λ_n(A)$, define \[ \mathcal{E}_p^+(A)=\sum_{λ_i > 0} λ_i^p(A), \quad \mathcal{E}_p^-(A)=\sum_{λ_i<0} |λ_i(A)|^p,\] to be the positive and the negative $p$-energy of $A$, respectively. In this note, first we show that if $A=[A_{ij}]_{i,j=1}^k$, where $A_{ii}$ are square matrices, then \[ \mathcal{E}_p^+(A)\geq \sum_{i=1}^{k} \mathcal{E}_p^+(A_{ii}), \quad \mathcal{E}_p^-(A)\geq \sum_{i=1}^{k} \mathcal{E}_p^-(A_{ii}),\] for any real number $p\geq 1$. We then apply the previous inequality to establish lower bounds for $p$-energy of the adjacency matrix of graphs.

math.CO↗

arXiv2025

A Counterexample to a Conjecture of Lovász

In 1975 Lovász conjectured that every $r$-partite, $r$-uniform hypergraph contains $r-1$ vertices whose deletion reduces the matching number. If true, this statement would imply a well-known conjecture of Ryser from 1971, which states that every $r$-partite, $r$-uniform hypergraph has a vertex cover of size at most $r-1$ times its matching number. When $r=2$, Ryser's conjecture is simply Kőnig's theorem, and the conjecture of Lovász is an immediate corollary. Ryser's conjecture for $r=3$ was proven by Aharoni in 2001, and remains open for all $r\geq 4$. Here we show that the conjecture of Lovász is false in the case $r=3$. Our counterexample is the line hypergraph of the Biggs-Smith graph, a highly symmetric cubic graph on 102 vertices.

math.CO↗

arXiv2025

Refinement of a conjecture on positive square energy of graphs

Let $G$ be a simple graph of order $n$ with eigenvalues $λ_1(G)\geq \cdots \geq λ_n(G)$. Define \[s^+(G)=\sum_{λ_i >0} λ_i^2(G), \quad s^-(G)=\sum_{λ_i<0} λ_i^2(G).\] It was conjectured by Elphick, Farber, Goldberg and Wocjan that for every connected graph $G$ of order $n$, $s^+(G) \ge n-1.$ We verify this conjecture for graphs with domination number at most 2. We then strengthen the conjecture as follows: if $G$ is a connected graph of order $n$ and size $m \geq n+1$, then $s^+(G) \geq n$. We prove this conjecture for claw-free graphs and graphs with diameter 2.

math.CO↗

arXiv2025

Three-edge-coloring (Tait coloring) cubic graphs on the torus: A proof of Grünbaum's conjecture

We prove that every cyclically 4-edge-connected cubic graph that can be embedded in the torus, with the exceptional graph class called "Petersen-like", is 3-edge-colorable. This means every (non-trivial) toroidal snark can be obtained from several copies of the Petersen graph using the dot product operation. The first two snarks in this family are the Petersen graph and one of Blanuša snarks; the rest are exposed by Vodopivec in 2008. This proves a strengthening of the well-known, long-standing conjecture of Grünbaum from 1968. This implies that a 2-connected cubic (multi)graph that can be embedded in the torus is not 3-edge-colorable if and only if it can be obtained from a dot product of copies of the Petersen graph by replacing its vertices with 2-edge-connected planar cubic (multi)graphs. Here, replacing a vertex $v$ in a cubic graph $G$ is the operation that takes a 2-connected planar cubic multigraph $H$ and one of its vertices $u$ of degree 3, unifying $G-v$ and $H-u$ and connecting the neighbors of $v$ in $G-v$ with the neighbors of $u$ in $H-u$ with a matching. This result is a highly nontrivial generalization of the Four Color Theorem, and its proof requires a combination of extensive computer verification and computer-free extension of existing proofs on colorability. An important consequence of this result is a very strong version of the Tutte 4-Flow Conjecture for toroidal graphs. We show that a 2-edge connected graph embedded in the torus admits a nowhere-zero 4-flow unless it is Petersen-like (in which case it does not admit nowhere-zero 4-flows). Observe that this is a vast strengthening over the Tutte 4-Flow Conjecture on the torus, which assumes that the graph does not contain the Petersen graph as a minor because almost all toroidal graphs contain the Petersen graph minor, but almost none are Petersen-like.

math.CO↗

arXiv2025

On a conjecture concerning 4-coloring of graphs with one crossing

We conjecture that every graph of minimum degree five with no separating triangles and drawn in the plane with one crossing is 4-colorable. In this paper, we use computer enumeration to show that this conjecture holds for all graphs with at most 28 vertices, explore the consequences of this conjecture and provide some insights on how it could be proved.

math.CO↗

arXiv2025

Random Embeddings of Graphs: The Expected Number of Faces in Most Graphs is Logarithmic

A random 2-cell embedding of a connected graph $G$ in some orientable surface is obtained by choosing a random local rotation around each vertex. Under this setup, the number of faces or the genus of the corresponding 2-cell embedding becomes a random variable. Random embeddings of two particular graph classes, those of a bouquet of $n$ loops and those of $n$ parallel edges connecting two vertices, have been extensively studied and are well-understood. However, little is known about more general graphs. The results of this paper explain why Monte Carlo methods cannot work for approximating the minimum genus of graphs. In his breakthrough work [Permutation-partition pairs, JCTB 1991], Stahl developed the foundation of "random topological graph theory". Most of his results have been unsurpassed until today. In our work, we analyze the expected number of faces of random embeddings (equivalently, the average genus) of a graph $G$. It was very recently shown that for any graph $G$, the expected number of faces is at most linear. We show that the actual expected number of faces $F(G)$ is almost always much smaller. In particular, we prove: 1) $\frac{1}{2}\ln n - 2 < \mathbb{E}[F(K_n)] \le 3.65 \ln n +o(1)$. 2) For random graphs $G(n,p)$ ($p=p(n)$), we have $\mathbb{E}[F(G(n,p))] \le \ln^2 n+\frac{1}{p}$. 3) For random models $B(n,Δ)$ containing only graphs, whose maximum degree is at most $Δ$, we obtain stronger bounds by showing that the expected number of faces is $Θ(\log n)$.

math.CO↗

arXiv2024

Exact Algorithms for Clustered Planarity with Linear Saturators

We study Clustered Planarity with Linear Saturators, which is the problem of augmenting an $n$-vertex planar graph whose vertices are partitioned into independent sets (called clusters) with paths - one for each cluster - that connect all the vertices in each cluster while maintaining planarity. We show that the problem can be solved in time $2^{O(n)}$ for both the variable and fixed embedding case. Moreover, we show that it can be solved in subexponential time $2^{O(\sqrt{n}\log n)}$ in the fixed embedding case if additionally the input graph is connected. The latter time complexity is tight under the Exponential-Time Hypothesis. We also show that $n$ can be replaced with the vertex cover number of the input graph by providing a linear (resp. polynomial) kernel for the variable-embedding (resp. fixed-embedding) case; these results contrast the NP-hardness of the problem on graphs of bounded treewidth (and even on trees). Finally, we complement known lower bounds for the problem by showing that Clustered Planarity with Linear Saturators is NP-hard even when the number of clusters is at most $3$, thus excluding the algorithmic use of the number of clusters as a parameter.

cs.DS↗

arXiv2024

A Linear Lower Bound for the Square Energy of Graphs

Let $G$ be a graph of order $n$ with eigenvalues $λ_1 \geq \cdots \geqλ_n$. Let \[s^+(G)=\sum_{λ_i>0} λ_i^2, \qquad s^-(G)=\sum_{λ_i<0} λ_i^2.\] The smaller value, $s(G)=\min\{s^+(G), s^-(G)\}$ is called the \emph{square energy} of $G$. In 2016, Elphick, Farber, Goldberg and Wocjan conjectured that for every connected graph $G$ of order $n$, $s(G)\geq n-1.$ No linear bound for $s(G)$ in terms of $n$ is known. Let $H_1, \ldots, H_k$ be disjoint vertex-induced subgraphs of $G$. In this note, we prove that \[s^+(G)\geq\sum_{i=1}^{k} s^+(H_i) \quad \text{ and } \quad s^-(G)\geq\sum_{i=1}^{k} s^-(H_i),\] which implies that $s(G)\geq \frac{3n}{4}$ for every connected graph $G$ of order $n\ge 4$.

math.CO↗

arXiv2024

Bipartite graphs are $(\frac{4}{5}-\varepsilon) \fracΔ{\log Δ}$-choosable

Alon and Krivelevich conjectured that if $G$ is a bipartite graph of maximum degree $Δ$, then the choosability (or list chromatic number) of $G$ satisfies $χ_{\ell}(G) = O \left ( \log Δ\right )$. Currently, the best known upper bound for $χ_{\ell}(G)$ is $(1 + o(1)) \fracΔ{\log Δ}$, which also holds for the much larger class of triangle-free graphs. We prove that for $\varepsilon = 10^{-3}$, every bipartite graph $G$ of sufficiently large maximum degree $Δ$ satisfies $χ_{\ell}(G) < (\frac{4}{5} -\varepsilon) \fracΔ{\log Δ}$. This improved upper bound suggests that list coloring is fundamentally different for bipartite graphs than for triangle-free graphs and hence gives a step toward solving the conjecture of Alon and Krivelevich.

math.CO↗

arXiv2024

On the second largest adjacency eigenvalue of trees with given diameter

For a graph $G$, let $λ_2(G)$ denote the second largest eigenvalue of the adjacency matrix of $G$. We determine the extremal trees with maximum/minimum adjacency eigenvalue $λ_2$ in the class $\mathcal{T}(n,d)$ of $n$-vertex trees with diameter $d$. This contributes to the literature on $λ_2$-extremization over different graph families. We also revisit the notion of the spectral center of a tree and the proof of $λ_2$ maximization over trees.

math.CO↗

arXiv2024

Efficient polynomial-time approximation scheme for the genus of dense graphs

The main results of this paper provide an Efficient Polynomial-Time Approximation Scheme (EPTAS) for approximating the genus (and non-orientable genus) of dense graphs. By dense we mean that $|E(G)|\ge α|V(G)|^2$ for some fixed $α>0$. While a constant factor approximation is trivial for this class of graphs, approximations with factor arbitrarily close to 1 need a sophisticated algorithm and complicated mathematical justification. More precisely, we provide an algorithm that for a given (dense) graph $G$ of order $n$ and given $\varepsilon>0$, returns an integer $g$ such that $G$ has an embedding into a surface of genus $g$, and this is $\varepsilon$-close to a minimum genus embedding in the sense that the minimum genus $\mathsf{g}(G)$ of $G$ satisfies: $\mathsf{g}(G)\le g\le (1+\varepsilon)\mathsf{g}(G)$. The running time of the algorithm is $O(f(\varepsilon)\,n^2)$, where $f(\cdot)$ is an explicit function. Next, we extend this algorithm to also output an embedding (rotation system) whose genus is $g$. This second algorithm is an Efficient Polynomial-time Randomized Approximation Scheme (EPRAS) and runs in time $O(f_1(\varepsilon)\,n^2)$.

math.CO↗

arXiv2024

Clustered Colouring of Graph Products

A colouring of a graph $G$ has clustering $k$ if the maximum number of vertices in a monochromatic component equals $k$. Motivated by recent results showing that many natural graph classes are subgraphs of the strong product of a graph with bounded treewidth and a path, this paper studies clustered colouring of strong products of two bounded treewidth graphs, where none, one, or both graphs have bounded degree. For example, in the case of two colours, if $n$ is the number of vertices in the product, then we show that clustering $Θ(n^{2/3})$ is best possible, even if one of the graphs is a path. However, if both graphs have bounded degree, then clustering $Θ(n^{1/2})$ is best possible. With three colours, if one of the graphs has bounded degree, then we show that clustering $Θ(n^{3/7})$ is best possible. However, if neither graph has bounded degree, then clustering $Ω(n^{1/2})$ is necessary. More general bounds for any given number of colours are also presented.

math.CO↗

arXiv2024

List-avoiding orientations

Given a graph $G$ with a set $F(v)$ of forbidden values at each $v \in V(G)$, an $F$-avoiding orientation of $G$ is an orientation in which $deg^+(v) \not \in F(v)$ for each vertex $v$. Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati conjectured that if $|F(v)| < \frac{1}{2} deg(v)$ for each $v \in V(G)$, then $G$ has an $F$-avoiding orientation, and they showed that this statement is true when $\frac{1}{2}$ is replaced by $\frac{1}{4}$. In this paper, we take a step toward this conjecture by proving that if $|F(v)| < \lfloor \frac{1}{3} deg(v) \rfloor$ for each vertex $v$, then $G$ has an $F$-avoiding orientation. Furthermore, we show that if the maximum degree of $G$ is subexponential in terms of the minimum degree, then this coefficient of $\frac{1}{3}$ can be increased to $\sqrt{2} - 1 - o(1) \approx 0.414$. Our main tool is a new sufficient condition for the existence of an $F$-avoiding orientation based on the Combinatorial Nullstellensatz of Alon and Tarsi.

math.CO↗

arXiv2024

Guarding isometric subgraphs and Cops and Robber in planar graphs

In the game of Cops and Robbers, one of the most useful results is that an isometric path in a graph can be guarded by one cop. In this paper, we introduce the concept of wide shadow in a subgraph, and use it to characterize all 1-guardable graphs. As an application, we show that 3 cops can capture a robber in any planar graph with the added restriction that at most two cops can move simultaneously, proving a conjecture of Yang and strengthening a classical result of Aigner and Fromme.

math.CO↗

arXiv2024

Three-edge-coloring projective planar cubic graphs: A generalization of the Four Color Theorem

We prove that every cyclically 4-edge-connected cubic graph that can be embedded in the projective plane, with the single exception of the Petersen graph, is 3-edge-colorable. In other words, the only (non-trivial) snark that can be embedded in the projective plane is the Petersen graph. This implies that a 2-connected cubic (multi)graph that can be embedded in the projective plane is not 3-edge-colorable if and only if it can be obtained from the Petersen graph by replacing each vertex by a 2-edge-connected planar cubic (multi)graph. This result is a nontrivial generalization of the Four Color Theorem, and its proof requires a combination of extensive computer verification and computer-free extension of existing proofs on colorability. An unexpected consequence of this result is a coloring-flow duality statement for the projective plane: A cubic graph embedded in the projective plane is 3-edge-colorable if and only if its dual multigraph is 5-vertex-colorable. Moreover, we show that a 2-edge connected graph embedded in the projective plane admits a nowhere-zero 4-flow unless it is Peteren-like (in which case it does not admit nowhere-zero 4-flows). This proves a strengthening of the Tutte 4-flow conjecture for graphs on the projective plane. Some of our proofs require extensive computer verification. The necessary source codes, together with the input and output files and the complete set of more than 6000 reducible configurations are available on Github (https://github.com/edge-coloring) which can be considered as an Addendum to this paper. Moreover, we provide pseudocodes for all our computer verifications.

math.CO↗

arXiv2024

Catching a robber on a random $k$-uniform hypergraph

The game of \emph{Cops and Robber} is usually played on a graph, where a group of cops attempt to catch a robber moving along the edges of the graph. The \emph{cop number} of a graph is the minimum number of cops required to win the game. An important conjecture in this area, due to Meyniel, states that the cop number of an $n$-vertex connected graph is $O(\sqrt{n})$. In 2016, Prałat and Wormald [Meyniel's conjecture holds for random graphs, Random Structures Algorithms. 48 (2016), no. 2, 396-421. MR3449604] showed that this conjecture holds with high probability for random graphs above the connectedness threshold. Moreoever, Łuczak and Prałat [Chasing robbers on random graphs: Zigzag theorem, Random Structures Algorithms. 37 (2010), no. 4, 516-524. MR2760362] showed that on a $\log$-scale the cop number demonstrates a surprising \emph{zigzag} behaviour in dense regimes of the binomial random graph $G(n,p)$. In this paper, we consider the game of Cops and Robber on a hypergraph, where the players move along hyperedges instead of edges. We show that with high probability the cop number of the $k$-uniform binomial random hypergraph $G^k(n,p)$ is $O\left(\sqrt{\frac{n}{k}}\, \log n \right)$ for a broad range of parameters $p$ and $k$ and that on a $\log$-scale our upper bound on the cop number arises as the minimum of \emph{two} complementary zigzag curves, as opposed to the case of $G(n,p)$. Furthermore, we conjecture that the cop number of a connected $k$-uniform hypergraph on $n$ vertices is $O\left(\sqrt{\frac{n}{k}}\,\right)$.

math.CO↗