arXiv subjects

Daniel A. Quiroz

Publications and source records attributed to Daniel A. Quiroz.

At least 19 recordsLinked to original sources

arXiv2026

Odd minors or odd immersions in graphs with independence number two

Kühn, Sauermann, Steiner and Wigderson recently disproved the Odd Hadwiger Conjecture, even for graphs with independence number 2. For this class of graphs the conjecture is known to be equivalent to the following: every $n$-vertex graph $G$ with independence number 2 contains $K_{\lceil \frac n2 \rceil}$ as an odd minor. While this does not hold, we prove that every graph $G$ with independence number 2 contains $K_{\lceil \frac n2 \rceil}$ as an odd minor or as a totally odd immersion.

math.CO↗

arXiv2026

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.

math.CO↗

arXiv2026

Odd Hadwiger number and graph products

The Odd Hadwiger number of a graph $G$ is the largest integer $r$ such that $G$ has a clique of size $r$ as an odd minor. In this paper, we investigate how large is the Odd Hadwiger number of the product of two graphs, when considering any of the four standard graph products: Cartesian, direct, lexicographic, strong. We provide an optimal lower bound in the cases of the strong and lexicographic products.

math.CO↗

arXiv2026

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}, $χ_b(\hat{G})$, 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 $χ(G)=χ_b(\tilde{G})$, 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 $(G, σ)$ 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↗

arXiv2026

Homomorphism counting for immersion-closed classes is not isomorphism

Lovász proved that two graphs $G$ and $H$ are isomorphic if $\hom(K,G) = \hom(K,H)$ for all graphs $K$, where $\hom(G_1,G_2)$ denotes the number of homomorphisms from $G_1$ to $G_2$. Dvořák showed that it suffices to count homomorphisms from all $2$-degenerate graphs $K$. On the other hand, for several interesting graph classes $\mathcal{M}$, it has been shown that there exist non-isomorphic graphs $G$ and $H$ such that $\hom(K,G)=\hom(K,H)$ for all $K\in \mathcal{M}$. Most such classes are minor-closed and Roberson conjectured that every proper minor-closed graph class $\mathcal{M}$ has the property that there exist non-isomorphic graphs that are indistinguishable by homomorphism counts from $\mathcal{M}$. There has been an effort to prove Roberson's conjecture as it is believed that minor-closed classes play a special role in demarcating the graph classes that satisfy this property. We show that this special role, if it exists, must be shared, by proving an analogue of Roberson's conjecture for a rich family of non-minor-closed classes. Namely, we prove that for any proper immersion-closed graph class $\mathcal{M}$, there exist non-isomorphic graphs $G$ and $H$ such that $\hom(K,G) = \hom(K,H)$ for all $K \in \mathcal{M}$. This extends a result of Roberson on homomorphism indistinguishability over bounded degree graphs, and cannot be extended in the natural way by replacing immersions with topological minors due to a result of Neuen and Seppelt. Our main result is obtained as a consequence of a colouring result which may be of independent interest: Every graph with a $\oddbound$-oddomorphism admits a $K_{t}$-immersion.

math.CO↗

arXiv2025

Boundedness for proper conflict-free and odd colorings

The proper conflict-free chromatic number, $χ_{pcf}(G)$, of a graph $G$ is the least $k$ such that $G$ has a proper $k$-coloring in which for each non-isolated vertex there is a color appearing exactly once among its neighbors. The proper odd chromatic number, $χ_{o}(G)$, of $G$ is the least $k$ such that $G$ has a proper coloring in which for every non-isolated vertex there is a color appearing an odd number of times among its neighbors. We say that a graph class $\mathcal{G}$ is $χ_{pcf}$-bounded ($χ_{o}$-bounded) if there is a function $f$ such that $χ_{pcf}(G) \leq f(χ(G))$ ($χ_{o}(G) \leq f(χ(G))$) for every $G \in \mathcal{G}$. Caro et al. (2022) asked for classes that are linearly $χ_{pcf}$-bounded ($χ_{pcf}$-bounded), and as a starting point, they showed that every claw-free graph $G$ satisfies $χ_{pcf}(G) \le 2Δ(G)+1$, which implies $χ_{pcf}(G) \le 4χ(G)+1$. In this paper, we improve the bound for claw-free graphs to a nearly tight bound by showing that such a graph $G$ satisfies $χ_{pcf}(G) \le Δ(G)+6$, and even $χ_{pcf}(G) \le Δ(G)+4$ if it is a quasi-line graph. These results also give evidence for a conjecture by Caro et al. Moreover, we show that convex-round graphs and permutation graphs are linearly $χ_{pcf}$-bounded. For these last two results, we prove a lemma that reduces the problem of deciding if a hereditary class is linearly $χ_{pcf}$-bounded to deciding if the bipartite graphs in the class are $χ_{pcf}$-bounded by an absolute constant. This lemma complements a theorem of Liu (2022) and motivates us to study boundedness in bipartite graphs. In particular, we show that biconvex bipartite graphs are $χ_{pcf}$-bounded while convex bipartite graphs are not even $χ_o$-bounded, and exhibit a class of bipartite circle graphs that is linearly $χ_o$-bounded but not $χ_{pcf}$-bounded.

math.CO↗

arXiv2025

Profile and neighbourhood complexity of graphs excluding a minor and tree-structured graphs

The \emph{$r$-neighbourhood complexity} of a graph $G$ is the function counting, for a given integer $k$, the largest possible number, over all vertex-subsets $A$ of size $k$, of subsets of $A$ realized as the intersection between the $r$-neighbourhood of some vertex and $A$. A~refinement of this notion is the \emph{$r$-profile complexity}, that counts the maximum number of distinct distance-vectors from any vertex to the vertices of $A$, ignoring distances larger than~$r$. Typically, in structured graph classes such as graphs of bounded VC-dimension or chordal graphs, these functions are bounded, leading to insights into their structural properties and efficient algorithms. We improve existing bounds on the $r$-profile complexity (and thus on the $r$-neighbourhood complexity) for graphs in several structured graph classes. We show that the $r$-profile complexity of graphs excluding $K_h$ as a minor is in $O_h(r^{3h-3}k)$. For graphs of treewidth at most~$t$, we give a bound in $O_t(r^{t+1}k)$, which is tight up to a function of~$t$ as a factor. These bounds improve results of Joret and Rambaud and answer a question of their paper [Combinatorica, 2024]. We also apply our methods to other classes of bounded expansion such as graphs excluding a fixed complete graph as a subdivision. For outerplanar graphs, we can improve our treewidth bound by a factor of $r$ and conjecture that a similar improvement holds for graphs with bounded simple treewidth. For graphs of treelength at most~$\ell$, we give the upper bound of $O(k(r^2(\ell+1)^k))$, which we improve to $O\left (k\cdot (r 2^k + r^2k^2) \right)$ in the case of chordal graphs and $O(k^2r)$ for interval graphs. Our bounds also imply relations between the order, diameter and metric dimension of graphs in these classes, improving results from [Beaudou et al., SIDMA 2017].

cs.DM↗

arXiv2025

Totally odd immersions of complete graphs in graph products

For a graph $G$, let $im(G)$ denote the maximum integer $t$ such that $G$ contains $K_t$ as an immersion. A recent paper of Collins, Heenehan, and McDonald (2023) studied the behaviour of this parameter under graph products, asking how large can $im(G\ast H)$ be in terms of $im(G)$ and $im(H)$, when $\ast$ is one of the four standard graph products. We consider a similar question for the parameter $toi(G)$ which denotes the maximum integer $t$ such that $G$ contains $K_t$ as a totally odd immersion. As an application, we obtain that no minimum counterexample to the immersion-analogue of the Odd Hadwiger Conjecture can be obtained from the Cartesian, direct (tensor), lexicographic or strong product of graphs.

math.CO↗

arXiv2025

Totally odd subdivisions in Kneser graphs

As evidence for the Odd Hadwiger Conjecture, Simonyi and Zsbán (2010) showed that every Kneser graph $G$ with large enough order (compared to $χ(G)$) contains a totally odd subdivision of $K_{χ(G)}$. A recent result of Steiner (2024), shows that every Schriver graph, and thus every Kneser graph, satisfies the Odd Hadwiger Conjecture, that is, it contains $K_{χ(G)}$ as an odd minor. We strengthen these results for Kneser graphs in two ways. We show that for every $t\ge 8$, there are $t$-chromatic Kneser graphs that contain arbitrarily large complete totally odd subdivisions (and thus, odd minors). We also show that every Kneser graph contains a totally odd subdivision of $K_{χ(G)}$. Kneser graphs are the prime example of graphs having chromatic number equal to its topological lower bounds. Motivated by our main results, we also study totally odd immersions on graphs with this property, proving, in particular, that if the chromatic number of $G$ is equal to any of its topological lower bounds, then $G$ contains a totally odd immersion of $K_{\lfloor χ(G)/2 \rfloor +1}$. This gives evidence for the immersion-analogue of the Odd Hadwiger Conjecture.

math.CO↗

arXiv2024

Biclique immersions in graphs with independence number 2

The analogue of Hadwiger's conjecture for the immersion relation states that every graph $G$ contains an immersion of $K_{χ(G)}$. For graphs with independence number 2, this is equivalent to stating that every such $n$-vertex graph contains an immersion of $K_{\lceil n/2 \rceil}$. We show that every $n$-vertex graph with independence number 2 contains every complete bipartite graph on $\lceil n/2 \rceil$ vertices as an immersion.

math.CO↗

arXiv2024

Subchromatic numbers of powers of graphs with excluded minors

A $k$-subcolouring of a graph $G$ is a function $f:V(G) \to \{0,\ldots,k-1\}$ such that the set of vertices coloured $i$ induce a disjoint union of cliques. The subchromatic number, $χ_{\textrm{sub}}(G)$, is the minimum $k$ such that $G$ admits a $k$-subcolouring. Nešetřil, Ossona de Mendez, Pilipczuk, and Zhu (2020), recently raised the problem of finding tight upper bounds for $χ_{\textrm{sub}}(G^2)$ when $G$ is planar. We show that $χ_{\textrm{sub}}(G^2)\le 43$ when $G$ is planar, improving their bound of 135. We give even better bounds when the planar graph $G$ has larger girth. Moreover, we show that $χ_{\textrm{sub}}(G^{3})\le 95$, improving the previous bound of 364. For these we adapt some recent techniques of Almulhim and Kierstead (2022), while also extending the decompositions of triangulated planar graphs of Van den Heuvel, Ossona de Mendez, Quiroz, Rabinovich and Siebertz (2017), to planar graphs of arbitrary girth. Note that these decompositions are the precursors of the graph product structure theorem of planar graphs. We give improved bounds for $χ_{\textrm{sub}}(G^p)$ for all $p$, whenever $G$ has bounded treewidth, bounded simple treewidth, bounded genus, or excludes a clique or biclique as a minor. For this we introduce a family of parameters which form a gradation between the strong and the weak colouring numbers. We give upper bounds for these parameters for graphs coming from such classes. Finally, we give a 2-approximation algorithm for the subchromatic number of graphs coming from any fixed class with bounded layered cliquewidth. In particular, this implies a 2-approximation algorithm for the subchromatic number of powers $G^p$ of graphs coming from any fixed class with bounded layered treewidth (such as the class of planar graphs). This algorithm works even if the power $p$ and the graph $G$ is unknown.

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.

math.CO↗

arXiv2023

Totally odd immersions in line graphs

The immersion-analogue of Hadwiger's Conjecture states that every graph $G$ contains an immersion of $K_{χ(G)}$. This conjecture has been recently strengthened in the following way: every graph $G$ contains a totally odd immersion of $K_{χ(G)}$. We prove this stronger conjecture for line graphs of constant-multiplicity multigraphs, thus extending a result of Guyer and McDonald.

math.CO↗

arXiv2022

The treewidth and pathwidth of graph unions

Given two $n$-vertex graphs $G_1$ and $G_2$ of bounded treewidth, is there an $n$-vertex graph $G$ of bounded treewidth having subgraphs isomorphic to $G_1$ and $G_2$? Our main result is a negative answer to this question, in a strong sense: we show that the answer is no even if $G_1$ is a binary tree and $G_2$ is a ternary tree. We also provide an extensive study of cases where such `gluing' is possible. In particular, we prove that if $G_1$ has treewidth $k$ and $G_2$ has pathwidth $\ell$, then there is an $n$-vertex graph of treewidth at most $k + 3 \ell + 1$ containing both $G_1$ and $G_2$ as subgraphs.

math.CO↗

arXiv2022

Clique immersions and independence number

The analogue of Hadwiger's conjecture for the immersion order states that every graph $G$ contains $K_{χ(G)}$ as an immersion. If true, it would imply that every graph with $n$ vertices and independence number $α$ contains $K_{\lceil \frac nα\rceil}$ as an immersion. The best currently known bound for this conjecture is due to Gauthier, Le and Wollan, who recently proved that every graph $G$ contains an immersion of a clique on $\bigl\lceil \frac{χ(G)-4}{3.54}\bigr\rceil$ vertices. Their result implies that every $n$-vertex graph with independence number $α$ contains an immersion of a clique on $\bigl\lceil \frac{n}{3.54α}-1.13\bigr\rceil$ vertices. We improve on this result for all $α\ge 3$, by showing that every $n$-vertex graph with independence number $α\ge 3$ contains an immersion of a clique on $\bigl\lfloor \frac {n}{2.25 α- f(α)} \bigr\rfloor - 1$ vertices, where $f$ is a nonnegative function.

math.CO↗

arXiv2021

Universal arrays

A word on $q$ symbols is a sequence of letters from a fixed alphabet of size $q$. For an integer $k\ge 1$, we say that a word $w$ is $k$-universal if, given an arbitrary word of length $k$, one can obtain it by removing entries from $w$. It is easily seen that the minimum length of a $k$-universal word on $q$ symbols is exactly $qk$. We prove that almost every word of size $(1+o(1))c_qk$ is $k$-universal with high probability, where $c_q$ is an explicit constant whose value is roughly $q\log q$. Moreover, we show that the $k$-universality property for uniformly chosen words exhibits a sharp threshold. Finally, by extending techniques of Alon [Geometric and Functional Analysis 27 (2017), no. 1, 1--32], we give asymptotically tight bounds for every higher dimensional analogue of this problem.

math.CO↗

arXiv2021

Clique immersions in graphs of independence number two with certain forbidden subgraphs

The Lescure-Meyniel conjecture is the analogue of Hadwiger's conjecture for the immersion order. It states that every graph $G$ contains the complete graph $K_{χ(G)}$ as an immersion, and like its minor-order counterpart it is open even for graphs with independence number 2. We show that every graph $G$ with independence number $α(G)\ge 2$ and no hole of length between $4$ and $2α(G)$ satisfies this conjecture. In particular, every $C_4$-free graph $G$ with $α(G)= 2$ satisfies the Lescure-Meyniel conjecture. We give another generalisation of this corollary, as follows. Let $G$ and $H$ be graphs with independence number at most 2, such that $|V(H)|\le 4$. If $G$ is $H$-free, then $G$ satisfies the Lescure-Meyniel conjecture.

math.CO↗

arXiv2019

Colouring exact distance graphs of chordal graphs

For a graph $G=(V,E)$ and positive integer $p$, the exact distance-$p$ graph $G^{[\natural p]}$ 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 $p$. Recently, there has been an effort to obtain bounds on the chromatic number $χ(G^{[\natural p]})$ of exact distance-$p$ graphs for $G$ from certain classes of graphs. In particular, if a graph $G$ has tree-width $t$, it has been shown that $χ(G^{[\natural p]}) \in \mathcal{O}(p^{t-1})$ for odd $p$, and $χ(G^{[\natural p]}) \in \mathcal{O}(p^{t}Δ(G))$ for even $p$. We show that if $G$ is chordal and has tree-width $t$, then $χ(G^{[\natural p]}) \in \mathcal{O}(p\, t^2)$ for odd $p$, and $χ(G^{[\natural p]}) \in \mathcal{O}(p\, t^2 Δ(G))$ for even $p$. If we could show that for every graph $H$ of tree-width $t$ there is a chordal graph $G$ of tree-width $t$ which contains $H$ as an isometric subgraph (i.e., a distance preserving subgraph), then our results would extend to all graphs of tree-width $t$. While we cannot do this, we show that for every graph $H$ of genus $g$ there is a graph $G$ which is a triangulation of genus $g$ and contains $H$ as an isometric subgraph.

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