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Carl Feghali

Publications and source records attributed to Carl Feghali.

50 records · Page 3Linked to original sources
arXiv2019

Reconfiguration Graph for Vertex Colourings of Weakly Chordal Graphs

The reconfiguration graph $R_k(G)$ of the $k$-colourings of a graph $G$ contains as its vertex set the $k$-colourings of $G$ and two colourings are joined by an edge if they differ in colour on just one vertex of $G$. We show that for each $k \geq 3$ there is a $k$-colourable weakly chordal graph $G$ such that $R_{k+1}(G)$ is disconnected. We also introduce a subclass of $k$-colourable weakly chordal graphs which we call $k$-colourable compact graphs and show that for each $k$-colourable compact graph $G$ on $n$ vertices, $R_{k+1}(G)$ has diameter $O(n^2)$. We show that this class contains all $k$-colourable co-chordal graphs and when $k = 3$ all $3$-colourable $(P_5, \overline{P_5}, C_5)$-free graphs. We also mention some open problems.

math.CO↗
arXiv2019

Enclosings of Decompositions of Complete Multigraphs in $2$-Edge-Connected $r$-Factorizations

A decomposition of a multigraph $G$ is a partition of its edges into subgraphs $G(1), \ldots , G(k)$. It is called an $r$-factorization if every $G(i)$ is $r$-regular and spanning. If $G$ is a subgraph of $H$, a decomposition of $G$ is said to be enclosed in a decomposition of $H$ if, for every $1 \leq i \leq k$, $G(i)$ is a subgraph of $H(i)$. Feghali and Johnson gave necessary and sufficient conditions for a given decomposition of $λK_n$ to be enclosed in some $2$-edge-connected $r$-factorization of $μK_{m}$ for some range of values for the parameters $n$, $m$, $λ$, $μ$, $r$: $r=2$, $μ>λ$ and either $m \geq 2n-1$, or $m=2n-2$ and $μ= 2$ and $λ=1$, or $n=3$ and $m=4$. We generalize their result to every $r \geq 2$ and $m \geq 2n - 2$. We also give some sufficient conditions for enclosing a given decomposition of $λK_n$ in some $2$-edge-connected $r$-factorization of $μK_{m}$ for every $r \geq 3$ and $m = (2 - C)n$, where $C$ is a constant that depends only on $r$, $λ$ and~$μ$.

math.CO↗
arXiv2019

Reconfiguring 10-colourings of planar graphs

Let $k \geq 1$ be an integer. The reconfiguration graph $R_k(G)$ of the $k$-colourings of a graph~$G$ has as vertex set the set of all possible $k$-colourings of $G$ and two colourings are adjacent if they differ on exactly one vertex. A conjecture of Cereceda from 2007 asserts that for every integer $\ell \geq k + 2$ and $k$-degenerate graph $G$ on $n$ vertices, $R_{\ell}(G)$ has diameter $O(n^2)$. The conjecture has been verified only when $\ell \geq 2k + 1$. We give a simple proof that if $G$ is a planar graph on $n$ vertices, then $R_{10}(G)$ has diameter at most $n^2$. Since planar graphs are $5$-degenerate, this affirms Cereceda's conjecture for planar graphs in the case $\ell = 2k$.

math.CO↗
arXiv2019

Intersecting non-uniform families containing subfamilies

A family of sets is said to be intersecting if every pair of sets in the family have non-empty intersection. In this paper, we initiate the study of intersecting non-uniform families of sets of one of two sizes containing given subfamilies. For a set $X$ and integer $r$, let $\binom{X}{r}$ denote the family $\{A \subseteq X: |X| = r\}$. Let $a$, $b$, and $n$ be positive integers such that $a < b$. We determine the maximum size of an intersecting family in $\binom{[n]}{a} \cup \binom{[2n]}{b}$ whenever $n > b$. For $n$ sufficiently large, we also determine the maximum size of an intersecting family in $\binom{[2n]}{a} \cup \binom{[n+1, 3n]}{a} \cup \binom{[n] \cup [2n + 1, 3n]}{a} \cup \binom{[3n]}{b}$ whenever $3n > 2b$ and $b > a + 2$. Our results are, in some sense, best possible. Our methods include the use of Katona's shadow intersection theorem and a recent diversity theorem of Kupavskii and~Zakharov.

math.CO↗
arXiv2018

Towards Cereceda's conjecture for planar graphs

The reconfiguration graph $R_k(G)$ of the $k$-colourings of a graph $G$ has as vertex set the set of all possible $k$-colourings of $G$ and two colourings are adjacent if they differ on the colour of exactly one vertex. Cereceda conjectured ten years ago that, for every $k$-degenerate graph $G$ on $n$ vertices, $R_{k+2}(G)$ has diameter $\mathcal{O}({n^2})$. The conjecture is wide open, with a best known bound of $\mathcal{O}({k^n})$, even for planar graphs. We improve this bound for planar graphs to $2^{\mathcal{O}({\sqrt{n}})}$. Our proof can be transformed into an algorithm that runs in $2^{\mathcal{O}({\sqrt{n}})}$ time.

cs.DM↗
arXiv2017

Independent Feedback Vertex Sets for Graphs of Bounded Diameter

The Near-Bipartiteness problem is that of deciding whether or not the vertices of a graph can be partitioned into sets $A$ and $B$, where $A$ is an independent set and $B$ induces a forest. The set $A$ in such a partition is said to be an independent feedback vertex set. Yang and Yuan proved that Near-Bipartiteness is polynomial-time solvable for graphs of diameter 2 and NP-complete for graphs of diameter 4. We show that Near-Bipartiteness is NP-complete for graphs of diameter 3, resolving their open problem. We also generalise their result for diameter 2 by proving that even the problem of computing a minimum independent feedback vertex is polynomial-time solvable for graphs of diameter 2.

cs.DS↗
arXiv2017

Independent Feedback Vertex Set for $P_5$-free Graphs

The NP-complete problem Feedback Vertex Set is that of deciding whether or not it is possible, for a given integer $k\geq 0$, to delete at most $k$ vertices from a given graph so that what remains is a forest. The variant in which the deleted vertices must form an independent set is called Independent Feedback Vertex Set and is also NP-complete. In fact, even deciding if an independent feedback vertex set exists is NP-complete and this problem is closely related to the $3$-Colouring problem, or equivalently, to the problem of deciding whether or not a graph has an independent odd cycle transversal, that is, an independent set of vertices whose deletion makes the graph bipartite. We initiate a systematic study of the complexity of Independent Feedback Vertex Set for $H$-free graphs. We prove that it is NP-complete if $H$ contains a claw or cycle. Tamura, Ito and Zhou proved that it is polynomial-time solvable for $P_4$-free graphs. We show that it remains polynomial-time solvable for $P_5$-free graphs. We prove analogous results for the Independent Odd Cycle Transversal problem, which asks whether or not a graph has an independent odd cycle transversal of size at most $k$ for a given integer $k\geq 0$. Finally, in line with our underlying research aim, we compare the complexity of Independent Feedback Vertex Set for $H$-free graphs with the complexity of $3$-Colouring, Independent Odd Cycle Transversal and other related problems.

cs.DS↗
arXiv2016

On a conjecture of Mohar concerning Kempe equivalence of regular graphs

Let $G$ be a graph with a vertex colouring $α$. Let $a$ and $b$ be two colours. Then a connected component of the subgraph induced by those vertices coloured either $a$ or $b$ is known as a Kempe chain. A colouring of $G$ obtained from $α$ by swapping the colours on the vertices of a Kempe chain is said to have been obtained by a Kempe change. Two colourings of $G$ are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. A conjecture of Mohar (2007) asserts that, for $k \geq 3$, all $k$-colourings of a $k$-regular graph that is not complete are Kempe equivalent. It was later shown that all $3$-colourings of a cubic graph that is neither $K_4$ nor the triangular prism are Kempe equivalent. In this paper, we prove that the conjecture holds for each $k\geq 4$. We also report the implications of this result on the validity of the Wang-Swendsen-Kotecký algorithm for the antiferromagnetic Potts model at zero-temperature.

cs.DM↗
arXiv2016

Enclosings of Decompositions of Complete Multigraphs in 2-Factorizations

Let $k$, $λ$ and $μ$ be positive integers. A decomposition of a multigraph $ λG$ into edge-disjoint subgraphs $G_1, \ldots , G_k$ is said to be \emph{enclosed} by a decomposition of a multigraph $μH$ into edge-disjoint subgraphs $H_1, \ldots , H_k$ if $μ> λ$ and $G_i$ is a subgraph of $H_i$, $1 \leq i \leq k$. In this paper we initiate the study of when a decomposition can be enclosed by a decomposition that consists of spanning subgraphs. A decomposition of a graph is a 2-factorization if each subgraph is 2-regular and is Hamiltonian if each subgraph is a Hamiltonian cycle. Let $n$ and $m$ be positive integers. We give necessary and sufficient conditions for enclosing a decomposition of $λK_n$ in a $2$-factorization of $μK_{n+m}$ whenever $μ>λ$ and $m \geq n-2$. We also give necessary and sufficient conditions for enclosing a decomposition of $λK_n$ in a Hamiltonian decomposition of $μK_{n+m}$ whenever $μ> λ$ and $m \geq n-1$, or $μ> λ$, $n=3$ and $m=1$, or $μ= 2$, $λ=1$ and $m=n-2$.

math.CO↗
arXiv2016

Erdős-Ko-Rado Theorems for a Family of Trees

Given a graph $G$ and an integer $r\geq 1$, let $\mathcal{I}^{(r)}(G)$ denote the family of independent sets of size $r$ of $G$. For a vertex $v$ of $G$, let $\mathcal{I}^{(r)}_v(G)$ denote the family of independent sets of size $r$ that contain~$v$. This family is called an $r$-star and $v$ is the centre of the star. Then $G$ is said to be $r$-EKR if no pairwise intersecting subfamily of $\mathcal{I}^{(r)}(G)$ is bigger than the largest $r$-star, and if every maximum size pairwise intersecting subfamily of $\mathcal{I}^{(r)}(G)$ is an $r$-star, then $G$ is said to be strictly $r$-EKR. Let $μ(G)$ denote the minimum size of a maximal independent set of $G$. Holroyd and Talbot conjectured that if $2r \leq μ(G)$, then $G$ is $r$-EKR and strictly $r$-EKR if $2r < μ(G)$. An elongated claw is a tree in which one vertex is designated the root and no vertex other than the root has degree greater than 2. A depth-two claw is an elongated claw in which every vertex of degree~1 is at distance 2 from the root. We show that if $G$ is a depth-two claw, then $G$ is strictly $r$-EKR if $2r \leq μ(G)+1$, confirming the conjecture of Holroyd and Talbot for this family. We also show that if $G $ is an elongated claw with $n$ leaves and at least one leaf adjacent to the root, then $G$ is $r$-EKR if $2r \leq n$. Hurlbert and Kamat had conjectured that one can always find a largest $r$-star of a tree whose centre is a leaf. Baber and Borg have separately shown this to be false. We show that, moreover, for all $n \geq 2$, $d \geq 3$, there exists a positive integer $r$ such that there is a tree where the centre of the largest $r$-star is a vertex of degree $n$ at distance at least $d$ from every leaf.

math.CO↗
arXiv2015

Kempe Equivalence of Colourings of Cubic Graphs

Given a graph $G=(V,E)$ and a proper vertex colouring of $G$, a Kempe chain is a subset of $V$ that induces a maximal connected subgraph of $G$ in which every vertex has one of two colours. To make a Kempe change is to obtain one colouring from another by exchanging the colours of vertices in a Kempe chain. Two colourings are Kempe equivalent if each can be obtained from the other by a series of Kempe changes. A conjecture of Mohar asserts that, for $k \geq 3$, all $k$-colourings of $k$-regular graphs that are not complete are Kempe equivalent. We address the case $k=3$ by showing that all $3$-colourings of a cubic graph $G$ are Kempe equivalent unless $G$ is the complete graph $K_4$ or the triangular prism.

cs.DM↗
arXiv2015

A Reconfigurations Analogue of Brooks' Theorem and its Consequences

Let $G$ be a simple undirected graph on $n$ vertices with maximum degree~$Δ$. Brooks' Theorem states that $G$ has a $Δ$-colouring unless~$G$ is a complete graph, or a cycle with an odd number of vertices. To recolour $G$ is to obtain a new proper colouring by changing the colour of one vertex. We show an analogue of Brooks' Theorem by proving that from any $k$-colouring, $k>Δ$, a $Δ$-colouring of $G$ can be obtained by a sequence of $O(n^2)$ recolourings using only the original $k$ colours unless $G$ is a complete graph or a cycle with an odd number of vertices, or $k=Δ+1$, $G$ is $Δ$-regular and, for each vertex $v$ in $G$, no two neighbours of $v$ are coloured alike. We use this result to study the reconfiguration graph $R_k(G)$ of the $k$-colourings of $G$. The vertex set of $R_k(G)$ is the set of all possible $k$-colourings of $G$ and two colourings are adjacent if they differ on exactly one vertex. We prove that for $Δ\geq 3$, $R_{Δ+1}(G)$ consists of isolated vertices and at most one further component which has diameter $O(n^2)$. This result enables us to complete both a structural classification and an algorithmic classification for reconfigurations of colourings of graphs of bounded maximum degree.

cs.CC↗
arXiv2015

Partitioning a Graph into Disjoint Cliques and a Triangle-free Graph

A graph $G = (V, E)$ is \emph{partitionable} if there exists a partition $\{A, B\}$ of $V$ such that $A$ induces a disjoint union of cliques and $B$ induces a triangle-free graph. In this paper we investigate the computational complexity of deciding whether a graph is partitionable. The problem is known to be $\NP$-complete on arbitrary graphs. Here it is proved that if a graph $G$ is bull-free, planar, perfect, $K_4$-free or does not contain certain holes then deciding whether $G$ is partitionable is $\NP$-complete. This answers an open question posed by Thomass{é}, Trotignon and Vuškovi{ć}. In contrast a finite list of forbidden induced subgraphs is given for partitionable cographs.

cs.CC↗